 reserve i,j,n,k,l for Nat;
 reserve T,S,X,Y,Z for Subset of MC-wff;
 reserve p,q,r,t,F,H,G for Element of MC-wff;
 reserve s,U,V for MC-formula;
reserve f,g for FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:];
 reserve X,T for Subset of MC-wff;
 reserve F,G,H,p,q,r,t for Element of MC-wff;
 reserve s,h for MC-formula;
 reserve f for FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:];
 reserve i,j for Element of NAT;
 reserve F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,G for MC-formula;
 reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x for Element of MC-wff;
reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 for object;

theorem Th66:
  (X |-_IPC p & X c= Y) implies Y |-_IPC p
proof
assume
A1: X |-_IPC p & X c= Y; then
  p in CnIPC(X); then
  ex f st f is_a_proof_wrt_IPC X & Effect_IPC(f) = p by Th16; then
  consider g such that L: g is_a_proof_wrt_IPC X & Effect_IPC(g) = p;
    g is_a_proof_wrt_IPC Y & Effect_IPC(g) = p by L,A1,Th10; then
  p in CnIPC(Y) by Th16;
  hence thesis;
end;
