Theorem dvelimALT 1353

Description: Version of dvelim 1352 that doesn't use ax-10 966. (See dvelimfALT 1153 for a version that doesn't use ax-11 967.)

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-17 971	. . 3 |- (-. A.x x = y -> A.z -. A.x x = y)
2		hba1 1003	. . . . . 6 |- (A.x x = z -> A.xA.x x = z)
3		ax16ALT 1271	. . . . . 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 1110	. . . . 5 |- (A.x x = z -> ((z = y -> ph) -> A.x(z = y -> ph)))
7	6	a1d 12	. . . 4 |- (A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
8	2	hbn 1004	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
9		hba1 1003	. . . . . . . 8 |- (A.x x = y -> A.xA.x x = y)
10	9	hbn 1004	. . . . . . 7 |- (-. A.x x = y -> A.x -. A.x x = y)
11	8, 10	hban 1009	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = y) -> A.x(-. A.x x = z /\ -. A.x x = y))
12		ax-12 968	. . . . . . 7 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
13	12	imp 350	. . . . . 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 1110	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = y) -> ((z = y -> ph) -> A.x(z = y -> ph)))
16	15	ex 373	. . . 4 |- (-. A.x x = z -> (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph))))
17	7, 16	pm2.61i 126	. . 3 |- (-. A.x x = y -> ((z = y -> ph) -> A.x(z = y -> ph)))
18	1, 17	hbald 1113	. 2 |- (-. A.x x = y -> (A.z(z = y -> ph) -> A.xA.z(z = y -> ph)))
19		ax-17 971	. . 3 |- (ps -> A.zps)
20		dvelimALT.2	. . 3 |- (z = y -> (ph <-> ps))
21	19, 20	equsal 1151	. 2 |- (A.z(z = y -> ph) <-> ps)
22	21	albii 999	. 2 |- (A.xA.z(z = y -> ph) <-> A.xps)
23	18, 21, 22	3imtr3g 552	1 |- (-. A.x x = y -> (ps -> A.xps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

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