Theorem dvelimALT 1744

Description: Version of dvelim 1743 that doesn't use ax-10 1308. (See dvelimfALT 1514 for a version that doesn't use ax-11 1309.)

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 1317	. . 3 |- (-. A.x x = y -> A.z -. A.x x = y)
2		hba1 1350	. . . . . 6 |- (A.x x = z -> A.xA.x x = z)
3		ax16ALT 1648	. . . . . 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 1468	. . . . 5 |- (A.x x = z -> ((z = y -> ph) -> A.x(z = y -> ph)))
7	6	a1d 15	. . . 4 |- (A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
8	2	hbn 1351	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
9		hba1 1350	. . . . . . . 8 |- (A.x x = y -> A.xA.x x = y)
10	9	hbn 1351	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
11	8, 10	hban 1356	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> A.x(-. A.x x = z /\ -. A.x x = y))
12		ax-12 1310	. . . . . . 7 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
13	12	imp 377	. . . . . 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 1468	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> ((z = y -> ph) -> A.x(z = y -> ph)))
16	15	ex 402	. . . 4 |- (-. A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
17	7, 16	pm2.61i 140	. . 3 |- (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph)))
18	1, 17	hbald 1471	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
19		ax-17 1317	. . 3 |- (ps -> A.zps)
20		dvelimALT.2	. . 3 |- (z = y -> (ph <-> ps))
21	19, 20	equsal 1511	. 2 |- (A.z(z = y -> ph) <-> ps)
22	21	albii 1346	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
23	18, 21, 22	3imtr3g 611	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

Copyright terms: Public domain