Theorem sbc6 1957

Description: An equivalence for class substitution. (The proof was shortened by Eric Schmidt, 17-Jan-2007.)

Hypothesis

Ref	Expression
sbc6.1	|- A e. V

Assertion

Ref	Expression
sbc6	|- ([A / x]ph <-> A.x(x = A -> ph))

Distinct variable group:   x,A

Proof of Theorem sbc6

Step	Hyp	Ref	Expression
1		sbc6.1	. 2 |- A e. V
2		sbc6g 1955	. 2 |- (A e. V -> ([A / x]ph <-> A.x(x = A -> ph)))
3	1, 2	ax-mp 7	1 |- ([A / x]ph <-> A.x(x = A -> ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170  Vcvv 1811

This theorem is referenced by:  ralpr 2428

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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