Theorem spimt 1974

Description: Closed theorem form of spim 1975. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)

Assertion

Ref	Expression
spimt	|- (( F/xps /\ A.x(x = y -> (ph -> ps))) -> (A.xph -> ps))

Proof of Theorem spimt

Step	Hyp	Ref	Expression
1		nfnf1 1790	. . . . 5 |- F/x F/xps
2		nfa1 1788	. . . . 5 |- F/xA.xph
3	1, 2	nfan 1824	. . . 4 |- F/x( F/xps /\ A.xph)
4		sp 1747	. . . . . . 7 |- (A.xph -> ph)
5	4	adantl 452	. . . . . 6 |- (( F/xps /\ A.xph) -> ph)
6		nfr 1761	. . . . . . 7 |- ( F/xps -> (ps -> A.xps))
7	6	adantr 451	. . . . . 6 |- (( F/xps /\ A.xph) -> (ps -> A.xps))
8	5, 7	embantd 50	. . . . 5 |- (( F/xps /\ A.xph) -> ((ph -> ps) -> A.xps))
9	8	imim2d 48	. . . 4 |- (( F/xps /\ A.xph) -> ((x = y -> (ph -> ps)) -> (x = y -> A.xps)))
10	3, 9	alimd 1764	. . 3 |- (( F/xps /\ A.xph) -> (A.x(x = y -> (ph -> ps)) -> A.x(x = y -> A.xps)))
11	10	impancom 427	. 2 |- (( F/xps /\ A.x(x = y -> (ph -> ps))) -> (A.xph -> A.x(x = y -> A.xps)))
12		ax9o 1950	. 2 |- (A.x(x = y -> A.xps) -> ps)
13	11, 12	syl6 29	1 |- (( F/xps /\ A.x(x = y -> (ph -> ps))) -> (A.xph -> ps))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   F/wnf 1544

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  spim  1975  equveli  1988