Theorem sbc5 1963

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 1961	. 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 960   e. wcel 962  E.wex 984  [wsbc 1176  Vcvv 1818

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-v 1819  df-sbc 1949

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