Theorem dvelimfALT 1514

Description: Proof of dvelimf 1623 without using ax-11 1309. See dvelimALT 1744 for a proof (of the distinct variable version dvelim 1743) that doesn't require ax-10 1308.

Hypotheses

Ref	Expression
dvelimfALT.1	|- (ph -> A.xph)
dvelimfALT.2	|- (ps -> A.zps)
dvelimfALT.3	|- (z = y -> (ph <-> ps))

Assertion

Ref	Expression
dvelimfALT	|- (-. A.x x = y -> (ps -> A.xps))

Proof of Theorem dvelimfALT

Step	Hyp	Ref	Expression
1		ax-10o 1500	. . . . . 6 |- (A.z z = x -> (A.zA.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
2	1	alequcoms 1503	. . . . 5 |- (A.x x = z -> (A.zA.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
3		hba1 1350	. . . . 5 |- (A.z(z = y -> ph) -> A.zA.z(z = y -> ph))
4	2, 3	syl5 20	. . . 4 |- (A.x x = z -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
5	4	a1d 15	. . 3 |- (A.x x = z -> (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph))))
6		hbnae 1507	. . . . . 6 |- (-. A.x x = z -> A.z -. A.x x = z)
7		hbnae 1507	. . . . . 6 |- (-. A.x x = y -> A.z -. A.x x = y)
8	6, 7	hban 1356	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> A.z(-. A.x x = z /\ -. A.x x = y))
9		hbnae 1507	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
10		hbnae 1507	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
11	9, 10	hban 1356	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> A.x(-. A.x x = z /\ -. A.x x = y))
12		ax-12 1310	. . . . . . 7 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
13	12	imp 377	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (z = y -> A.x z = y))
14		dvelimfALT.1	. . . . . . 7 |- (ph -> A.xph)
15	14	a1i 8	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (ph -> A.xph))
16	11, 13, 15	hbimd 1468	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> ((z = y -> ph) -> A.x(z = y -> ph)))
17	8, 16	hbald 1471	. . . 4 |- ((-. A.x x = z /\ -. A.x x = y) -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
18	17	ex 402	. . 3 |- (-. A.x x = z -> (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph))))
19	5, 18	pm2.61i 140	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
20		dvelimfALT.2	. . 3 |- (ps -> A.zps)
21		dvelimfALT.3	. . 3 |- (z = y -> (ph <-> ps))
22	20, 21	equsal 1511	. 2 |- (A.z(z = y -> ph) <-> ps)
23	22	albii 1346	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
24	19, 22, 23	3imtr3g 611	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298

This theorem is referenced by:  dveeq2 1582  dveeq2ALT 1583  dveeq1 1745  dveeq1ALT 1746  dveel1 1747  dveel2 1748  ax15 1750  dveel2ALT 1753  ax11el 1755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500

This theorem depends on definitions:  df-bi 164  df-an 242

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