Theorem sbcth2 3051

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 2530	. 2 |- A. x e. B ph
3		rspsbc 3046	. 2 |- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )
4	2, 3	mpi 15	1 |- ( A e. B -> [. A / x ]. ph )

Syntax hints:   -> wi 4   e. wcel 2148   A.wral 2455   [.wsbc 2963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2740  df-sbc 2964

This theorem is referenced by: (None)