Theorem dvelimfALT2 16366

Description: Proof of dvelimf 1623 using dveeq2 1582 instead of ax-12 1310. This shows that ax-12 1310 could be replaced by dveeq2 1582.

Hypotheses

Ref	Expression
dvelimfALT2.1	|- (ph -> A.xph)
dvelimfALT2.2	|- (ps -> A.zps)
dvelimfALT2.3	|- (z = y -> (ph <-> ps))

Assertion

Ref	Expression
dvelimfALT2	|- (-. A.x x = y -> (ps -> A.xps))

Distinct variable groups:   x,z   y,z

Proof of Theorem dvelimfALT2

Step	Hyp	Ref	Expression
1		ax-17 1317	. . 3 |- (-. A.x x = y -> A.z -. A.x x = y)
2		hbn1 1362	. . . 4 |- (-. A.x x = y -> A.x -. A.x x = y)
3		dveeq2 1582	. . . 4 |- (-. A.x x = y -> (z = y -> A.x z = y))
4		dvelimfALT2.1	. . . . 5 |- (ph -> A.xph)
5	4	a1i 8	. . . 4 |- (-. A.x x = y -> (ph -> A.xph))
6	2, 3, 5	hbimd 1468	. . 3 |- (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph)))
7	1, 6	hbald 1471	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
8		dvelimfALT2.2	. . 3 |- (ps -> A.zps)
9		dvelimfALT2.3	. . 3 |- (z = y -> (ph <-> ps))
10	8, 9	equsal 1511	. 2 |- (A.z(z = y -> ph) <-> ps)
11	10	albii 1346	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
12	7, 10, 11	3imtr3g 611	1 |- (-. A.x x = y -> (ps -> A.xps))

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

This theorem is referenced by:  ax12 16367

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-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  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

Copyright terms: Public domain