Theorem sbcth 2458

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 1305	. 2 |- A.xph
3		a4sbc 2457	. 2 |- (A e. B -> (A.xph -> [A / x]ph))
4	2, 3	mpi 55	1 |- (A e. B -> [A / x]ph)

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  [wsbc 1534

This theorem is referenced by:  sbcth2 2514  csbeq2i 2563  iota4an 5101  bnj895 12813

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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