Theorem spcgft 2931

Description: A closed version of spcgf 2934. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimgft.1	|- F/xps
spcimgft.2	|- F/_xA

Assertion

Ref	Expression
spcgft	|- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.xph -> ps)))

Proof of Theorem spcgft

Step	Hyp	Ref	Expression
1		bi1 178	. . . 4 |- ((ph <-> ps) -> (ph -> ps))
2	1	imim2i 13	. . 3 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ph -> ps)))
3	2	alimi 1559	. 2 |- (A.x(x = A -> (ph <-> ps)) -> A.x(x = A -> (ph -> ps)))
4		spcimgft.1	. . 3 |- F/xps
5		spcimgft.2	. . 3 |- F/_xA
6	4, 5	spcimgft 2930	. 2 |- (A.x(x = A -> (ph -> ps)) -> (A e. B -> (A.xph -> ps)))
7	3, 6	syl 15	1 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.xph -> ps)))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2476

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 2478  df-v 2861

This theorem is referenced by:  spcgf  2934  rspct  2948