Theorem sbcieg 1951

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 1808	. 2 |- (A e. B -> A e. V)
2		ax-17 968	. . . 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 1950	. 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 951   = wceq 953   e. wcel 955  [wsbc 1166  Vcvv 1802

This theorem is referenced by:  sbcie 1952

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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