Theorem sbc6 1964

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 1962	. 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 958   = wceq 960   e. wcel 962  [wsbc 1174  Vcvv 1818

This theorem is referenced by:  ralpr 2440

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-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-sbc 1949

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