Theorem sbcth 1917

Description: A substitution into a theorem remains true (when A is a set).

Hypothesis

Ref	Expression
sbcth.1	|- ph

Assertion

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

Proof of Theorem sbcth

Step	Hyp	Ref	Expression
1		sbcth.1	. . 3 |- ph
2	1	ax-gen 955	. 2 |- A.xph
3		a4sbc 1916	. 2 |- (A e. B -> (A.xph -> [A / x]ph))
4	2, 3	mpi 44	1 |- (A e. B -> [A / x]ph)

Syntax hints:   -> wi 3  A.wal 950   e. wcel 1105  [wsbc 1153

This theorem is referenced by:  sbcth2 1953  csbeq2i 1991

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955  ax-9 1102  ax-12 1104  ax-17 1190  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913

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