Theorem ax11v2 1585

Description: Recovery of ax11o 1587 from ax11v 1642 without using ax-11 1309. The hypothesis is even weaker than ax11v 1642, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1587.

Hypothesis

Ref	Expression
ax11v2.1	|- (x = z -> (ph -> A.x(x = z -> ph)))

Assertion

Ref	Expression
ax11v2	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

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

Proof of Theorem ax11v2

Step	Hyp	Ref	Expression
1		a9e 1483	. 2 |- E.z z = y
2		ax11v2.1	. . . . 5 |- (x = z -> (ph -> A.x(x = z -> ph)))
3		equequ2 1495	. . . . . . 7 |- (z = y -> (x = z <-> x = y))
4	3	adantl 424	. . . . . 6 |- ((-. A.x x = y /\ z = y) -> (x = z <-> x = y))
5		dveeq2 1582	. . . . . . . . 9 |- (-. A.x x = y -> (z = y -> A.x z = y))
6	5	imp 377	. . . . . . . 8 |- ((-. A.x x = y /\ z = y) -> A.x z = y)
7		hba1 1350	. . . . . . . . 9 |- (A.x z = y -> A.xA.x z = y)
8	3	imbi1d 675	. . . . . . . . . 10 |- (z = y -> ((x = z -> ph) <-> (x = y -> ph)))
9	8	a4s 1330	. . . . . . . . 9 |- (A.x z = y -> ((x = z -> ph) <-> (x = y -> ph)))
10	7, 9	albid 1459	. . . . . . . 8 |- (A.x z = y -> (A.x(x = z -> ph) <-> A.x(x = y -> ph)))
11	6, 10	syl 12	. . . . . . 7 |- ((-. A.x x = y /\ z = y) -> (A.x(x = z -> ph) <-> A.x(x = y -> ph)))
12	11	imbi2d 674	. . . . . 6 |- ((-. A.x x = y /\ z = y) -> ((ph -> A.x(x = z -> ph)) <-> (ph -> A.x(x = y -> ph))))
13	4, 12	imbi12d 688	. . . . 5 |- ((-. A.x x = y /\ z = y) -> ((x = z -> (ph -> A.x(x = z -> ph))) <-> (x = y -> (ph -> A.x(x = y -> ph)))))
14	2, 13	mpbii 210	. . . 4 |- ((-. A.x x = y /\ z = y) -> (x = y -> (ph -> A.x(x = y -> ph))))
15	14	ex 402	. . 3 |- (-. A.x x = y -> (z = y -> (x = y -> (ph -> A.x(x = y -> ph)))))
16	15	19.23adv 1584	. 2 |- (-. A.x x = y -> (E.z z = y -> (x = y -> (ph -> A.x(x = y -> ph)))))
17	1, 16	mpi 55	1 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326

This theorem is referenced by:  ax11a2 1586

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-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500

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

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