Theorem spcimgft 2931

Description: A closed version of spcimgf 2933. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem spcimgft

Step	Hyp	Ref	Expression
1		elex 2868	. 2 |- (A e. B -> A e. _V)
2		spcimgft.2	. . . . 5 |- F/_xA
3	2	issetf 2865	. . . 4 |- (A e. _V <-> E.x x = A)
4		exim 1575	. . . 4 |- (A.x(x = A -> (ph -> ps)) -> (E.x x = A -> E.x(ph -> ps)))
5	3, 4	syl5bi 208	. . 3 |- (A.x(x = A -> (ph -> ps)) -> (A e. _V -> E.x(ph -> ps)))
6		spcimgft.1	. . . 4 |- F/xps
7	6	19.36 1871	. . 3 |- (E.x(ph -> ps) <-> (A.xph -> ps))
8	5, 7	syl6ib 217	. 2 |- (A.x(x = A -> (ph -> ps)) -> (A e. _V -> (A.xph -> ps)))
9	1, 8	syl5 28	1 |- (A.x(x = A -> (ph -> ps)) -> (A e. B -> (A.xph -> ps)))

Syntax hints:   -> wi 4  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477  _Vcvv 2860

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-v 2862

This theorem is referenced by:  spcgft  2932  spcimgf  2933  spcimdv  2937