Theorem dvelimfALT2 1698

Description: Proof of dvelimf 1891 using dveeq2 1696 (shown as the last hypothesis) instead of ax-12 1402. This shows that ax-12 1402 could be replaced by dveeq2 1696 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.)

Hypotheses

Ref	Expression
dvelimfALT2.1	|- ( ph -> A. x ph )
dvelimfALT2.2	|- ( ps -> A. z ps )
dvelimfALT2.3	|- ( z = y -> ( ph <-> ps ) )
dvelimfALT2.4	|- ( -. A. x x = y -> ( z = y -> A. x z = y ) )

Assertion

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

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)

Proof of Theorem dvelimfALT2

Step	Hyp	Ref	Expression
1		ax-17 1419	. . 3 |- ( -. A. x x = y -> A. z -. A. x x = y )
2		hbn1 1542	. . . 4 |- ( -. A. x x = y -> A. x -. A. x x = y )
3		dvelimfALT2.4	. . . 4 |- ( -. A. x x = y -> ( z = y -> A. x z = y ) )
4		dvelimfALT2.1	. . . . 5 |- ( ph -> A. x ph )
5	4	a1i 9	. . . 4 |- ( -. A. x x = y -> ( ph -> A. x ph ) )
6	2, 3, 5	hbimd 1465	. . 3 |- ( -. A. x x = y -> ( ( z = y -> ph ) -> A. x ( z = y -> ph ) ) )
7	1, 6	hbald 1380	. 2 |- ( -. A. x x = y -> ( A. z ( z = y -> ph ) -> A. x A. z ( z = y -> ph ) ) )
8		dvelimfALT2.2	. . 3 |- ( ps -> A. z ps )
9		dvelimfALT2.3	. . 3 |- ( z = y -> ( ph <-> ps ) )
10	8, 9	equsalh 1614	. 2 |- ( A. z ( z = y -> ph ) <-> ps )
11	10	albii 1359	. 2 |- ( A. x A. z ( z = y -> ph ) <-> A. x ps )
12	7, 10, 11	3imtr3g 193	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by: (None)