Theorem rspct 2949

Description: A closed version of rspc 2950. (Contributed by Andrew Salmon, 6-Jun-2011.)

Hypothesis

Ref	Expression
rspct.1	|- F/xps

Assertion

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

Distinct variable groups:   x,A   x,B

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

Proof of Theorem rspct

Step	Hyp	Ref	Expression
1		df-ral 2620	. . . 4 |- (A.x e. B ph <-> A.x(x e. B -> ph))
2		eleq1 2413	. . . . . . . . . 10 |- (x = A -> (x e. B <-> A e. B))
3	2	adantr 451	. . . . . . . . 9 |- ((x = A /\ (ph <-> ps)) -> (x e. B <-> A e. B))
4		simpr 447	. . . . . . . . 9 |- ((x = A /\ (ph <-> ps)) -> (ph <-> ps))
5	3, 4	imbi12d 311	. . . . . . . 8 |- ((x = A /\ (ph <-> ps)) -> ((x e. B -> ph) <-> (A e. B -> ps)))
6	5	ex 423	. . . . . . 7 |- (x = A -> ((ph <-> ps) -> ((x e. B -> ph) <-> (A e. B -> ps))))
7	6	a2i 12	. . . . . 6 |- ((x = A -> (ph <-> ps)) -> (x = A -> ((x e. B -> ph) <-> (A e. B -> ps))))
8	7	alimi 1559	. . . . 5 |- (A.x(x = A -> (ph <-> ps)) -> A.x(x = A -> ((x e. B -> ph) <-> (A e. B -> ps))))
9		nfv 1619	. . . . . . 7 |- F/x A e. B
10		rspct.1	. . . . . . 7 |- F/xps
11	9, 10	nfim 1813	. . . . . 6 |- F/x(A e. B -> ps)
12		nfcv 2490	. . . . . 6 |- F/_xA
13	11, 12	spcgft 2932	. . . . 5 |- (A.x(x = A -> ((x e. B -> ph) <-> (A e. B -> ps))) -> (A e. B -> (A.x(x e. B -> ph) -> (A e. B -> ps))))
14	8, 13	syl 15	. . . 4 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.x(x e. B -> ph) -> (A e. B -> ps))))
15	1, 14	syl7bi 221	. . 3 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.x e. B ph -> (A e. B -> ps))))
16	15	com34 77	. 2 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A e. B -> (A.x e. B ph -> ps))))
17	16	pm2.43d 44	1 |- (A.x(x = A -> (ph <-> ps)) -> (A e. B -> (A.x e. B ph -> ps)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710  A.wral 2615

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

This theorem is referenced by: (None)