Theorem sbc5 1959

Description: An equivalence for class substitution.

Hypothesis

Ref	Expression
sbc5.1	|- A e. V

Assertion

Ref	Expression
sbc5	|- ([A / x]ph <-> E.x(x = A /\ ph))

Distinct variable group:   x,A

Proof of Theorem sbc5

Step	Hyp	Ref	Expression
1		sbc5.1	. 2 |- A e. V
2		sbc5g 1957	. 2 |- (A e. V -> ([A / x]ph <-> E.x(x = A /\ ph)))
3	1, 2	ax-mp 7	1 |- ([A / x]ph <-> E.x(x = A /\ ph))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  Vcvv 1814

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945

Copyright terms: Public domain