Theorem dvelimhw 2173

Description: Proof of dvelimh 2336 without using ax-13 2246 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.)

Hypotheses

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

Assertion

Ref	Expression
dvelimhw	|- ( -. 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 dvelimhw

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ z -. A. x x = y
2		equcom 1945	. . . . . 6 |- ( z = y <-> y = z )
3		nfna1 2029	. . . . . . 7 |- F/ x -. A. x x = y
4		dvelimhw.4	. . . . . . 7 |- ( -. A. x x = y -> ( y = z -> A. x y = z ) )
5	3, 4	nf5d 2118	. . . . . 6 |- ( -. A. x x = y -> F/ x y = z )
6	2, 5	nfxfrd 1780	. . . . 5 |- ( -. A. x x = y -> F/ x z = y )
7		dvelimhw.1	. . . . . . 7 |- ( ph -> A. x ph )
8	7	nf5i 2024	. . . . . 6 |- F/ x ph
9	8	a1i 11	. . . . 5 |- ( -. A. x x = y -> F/ x ph )
10	6, 9	nfimd 1823	. . . 4 |- ( -. A. x x = y -> F/ x ( z = y -> ph ) )
11	1, 10	nfald 2165	. . 3 |- ( -. A. x x = y -> F/ x A. z ( z = y -> ph ) )
12		dvelimhw.2	. . . . 5 |- ( ps -> A. z ps )
13		dvelimhw.3	. . . . 5 |- ( z = y -> ( ph <-> ps ) )
14	12, 13	equsalhw 2123	. . . 4 |- ( A. z ( z = y -> ph ) <-> ps )
15	14	nfbii 1778	. . 3 |- ( F/ x A. z ( z = y -> ph ) <-> F/ x ps )
16	11, 15	sylib 208	. 2 |- ( -. A. x x = y -> F/ x ps )
17	16	nf5rd 2066	1 |- ( -. A. x x = y -> ( ps -> A. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)