Theorem sbcth 1953

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 967	. 2 |- A.xph
3		a4sbc 1952	. 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 958   e. wcel 962  [wsbc 1174

This theorem is referenced by:  sbcth2 1992  csbeq2i 2031

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 967  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  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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