Theorem sbcth2 3130

Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
sbcth2.1	|- (x e. B -> ph)

Assertion

Ref	Expression
sbcth2	|- (A e. B -> [.A / x ].ph)

Distinct variable group:   x,B

Allowed substitution hints:   ph(x)   A(x)

Proof of Theorem sbcth2

Step	Hyp	Ref	Expression
1		sbcth2.1	. . 3 |- (x e. B -> ph)
2	1	rgen 2680	. 2 |- A.x e. B ph
3		rspsbc 3125	. 2 |- (A e. B -> (A.x e. B ph -> [.A / x ].ph))
4	2, 3	mpi 16	1 |- (A e. B -> [.A / x ].ph)

Syntax hints:   -> wi 4   e. wcel 1710  A.wral 2615   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)