Theorem sbcth 2735

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. V -> [A / x]ph)

Proof of Theorem sbcth

Step	Hyp	Ref	Expression
1		sbcth.1	. . 3 |- ph
2	1	ax-gen 1622	. 2 |- A.xph
3		a4sbc 2734	. 2 |- (A e. V -> (A.xph -> [A / x]ph))
4	2, 3	mpi 101	1 |- (A e. V -> [A / x]ph)

Syntax hints:   -> wi 3  A.wal 1613   e. wcel 1617  [wsbc 1843

This theorem is referenced by:  sbcth2 2780  csbeq2i 2826  iota4an 5275  bnj895 13742

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1622  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-ext 2152

This theorem depends on definitions:  df-bi 232  df-an 435  df-ex 1645  df-sb 1845  df-clab 2158  df-cleq 2163  df-clel 2166  df-v 2571  df-sbc 2731

Copyright terms: Public domain