Theorem dvelimh 1964

Description: Version of dvelim 2016 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)

Hypotheses

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

Assertion

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

Proof of Theorem dvelimh

Step	Hyp	Ref	Expression
1		hba1 1786	. . . . 5 |- (A.z(z = y -> ph) -> A.zA.z(z = y -> ph))
2		ax10o 1952	. . . . . 6 |- (A.z z = x -> (A.zA.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
3	2	aecoms 1947	. . . . 5 |- (A.x x = z -> (A.zA.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
4	1, 3	syl5 28	. . . 4 |- (A.x x = z -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
5	4	a1d 22	. . 3 |- (A.x x = z -> (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph))))
6		hbnae 1955	. . . . . 6 |- (-. A.x x = z -> A.z -. A.x x = z)
7		hbnae 1955	. . . . . 6 |- (-. A.x x = y -> A.z -. A.x x = y)
8	6, 7	hban 1828	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> A.z(-. A.x x = z /\ -. A.x x = y))
9		hbnae 1955	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
10		hbnae 1955	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
11	9, 10	hban 1828	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> A.x(-. A.x x = z /\ -. A.x x = y))
12		ax12o 1934	. . . . . . 7 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
13	12	imp 418	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (z = y -> A.x z = y))
14		dvelimh.1	. . . . . . 7 |- (ph -> A.xph)
15	14	a1i 10	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (ph -> A.xph))
16	11, 13, 15	hbimd 1815	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> ((z = y -> ph) -> A.x(z = y -> ph)))
17	8, 16	hbald 1740	. . . 4 |- ((-. A.x x = z /\ -. A.x x = y) -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
18	17	ex 423	. . 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 156	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
20		dvelimh.2	. . 3 |- (ps -> A.zps)
21		dvelimh.3	. . 3 |- (z = y -> (ph <-> ps))
22	20, 21	equsalh 1961	. 2 |- (A.z(z = y -> ph) <-> ps)
23	22	albii 1566	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
24	19, 22, 23	3imtr3g 260	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  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  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  dvelim  2016  dveeq1-o16  2188  dveel2ALT  2191