Theorem dvelimALT 2037

Description: Version of dvelim 2036 that doesn't use ax-10 1625. (See dvelimfALT 1823 for a version that doesn't use ax-11 1626.)

Hypotheses

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

Assertion

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

Distinct variable groups:   ps,z   x,z   y,z

Proof of Theorem dvelimALT

Step	Hyp	Ref	Expression
1		ax-17 1634	. . 3 |- (-. A.x x = y -> A.z -. A.x x = y)
2		hba1 1668	. . . . . 6 |- (A.x x = z -> A.xA.x x = z)
3		ax16ALT 1947	. . . . . 6 |- (A.x x = z -> (z = y -> A.x z = y))
4		dvelimALT.1	. . . . . . 7 |- (ph -> A.xph)
5	4	a1i 8	. . . . . 6 |- (A.x x = z -> (ph -> A.xph))
6	2, 3, 5	hbimd 1778	. . . . 5 |- (A.x x = z -> ((z = y -> ph) -> A.x(z = y -> ph)))
7	6	a1d 18	. . . 4 |- (A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
8	2	hbn 1669	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
9		hba1 1668	. . . . . . . 8 |- (A.x x = y -> A.xA.x x = y)
10	9	hbn 1669	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
11	8, 10	hban 1674	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> A.x(-. A.x x = z /\ -. A.x x = y))
12		ax-12 1627	. . . . . . 7 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
13	12	imp 489	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (z = y -> A.x z = y))
14	4	a1i 8	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> (ph -> A.xph))
15	11, 13, 14	hbimd 1778	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> ((z = y -> ph) -> A.x(z = y -> ph)))
16	15	ex 494	. . . 4 |- (-. A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
17	7, 16	pm2.61i 201	. . 3 |- (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph)))
18	1, 17	hbald 1782	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
19		ax-17 1634	. . 3 |- (ps -> A.zps)
20		dvelimALT.2	. . 3 |- (z = y -> (ph <-> ps))
21	19, 20	equsal 1821	. 2 |- (A.z(z = y -> ph) <-> ps)
22	21	albii 1664	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
23	18, 21, 22	3imtr3g 332	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 231   /\ wa 433  A.wal 1613   = wceq 1615

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-11 1626  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792

This theorem depends on definitions:  df-bi 232  df-an 435  df-ex 1645  df-sb 1845

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