Theorem sbc6 1954

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 1952	. 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 953   = wceq 955   e. wcel 957  [wsbc 1169  Vcvv 1808

This theorem is referenced by:  ralpr 2425

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939

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