Theorem dvelimhw 1849

Description: Proof of dvelimh 1964 without using ax-12 1925 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.)

Hypotheses

Ref	Expression
dvelimhw.1	|- (ph -> A.xph)
dvelimhw.2	|- (ps -> A.zps)
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.xps))

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		ax-17 1616	. . 3 |- (-. A.x x = y -> A.z -. A.x x = y)
2		hbn1 1730	. . . 4 |- (-. A.x x = y -> A.x -. A.x x = y)
3		equcomi 1679	. . . . 5 |- (z = y -> y = z)
4		dvelimhw.4	. . . . 5 |- (-. A.x x = y -> (y = z -> A.x y = z))
5		equcomi 1679	. . . . . 6 |- (y = z -> z = y)
6	5	alimi 1559	. . . . 5 |- (A.x y = z -> A.x z = y)
7	3, 4, 6	syl56 30	. . . 4 |- (-. A.x x = y -> (z = y -> A.x z = y))
8		dvelimhw.1	. . . . 5 |- (ph -> A.xph)
9	8	a1i 10	. . . 4 |- (-. A.x x = y -> (ph -> A.xph))
10	2, 7, 9	hbimd 1815	. . 3 |- (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph)))
11	1, 10	hbald 1740	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
12		dvelimhw.2	. . 3 |- (ps -> A.zps)
13		dvelimhw.3	. . 3 |- (z = y -> (ph <-> ps))
14	12, 13	equsalhw 1838	. 2 |- (A.z(z = y -> ph) <-> ps)
15	14	albii 1566	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
16	11, 14, 15	3imtr3g 260	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540

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

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax12olem6  1932