Theorem sbcieg 1961

Description: Conversion of implicit substitution to explicit class substitution.

Hypothesis

Ref	Expression
sbcieg.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
sbcieg	|- (A e. B -> ([A / x]ph <-> ps))

Distinct variable groups:   x,A   ps,x

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (A e. B -> A e. V)
2		ax-17 971	. . . 4 |- (ps -> A.xps)
3	2	a1i 8	. . 3 |- (A e. V -> (ps -> A.xps))
4		sbcieg.1	. . 3 |- (x = A -> (ph <-> ps))
5	3, 4	sbciegf 1960	. 2 |- (A e. V -> ([A / x]ph <-> ps))
6	1, 5	syl 10	1 |- (A e. B -> ([A / x]ph <-> ps))

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

This theorem is referenced by:  sbcie 1962

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-8 964  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-3an 777  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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