Theorem ax11v2 1257

Description: Recovery of ax11o 1259 from ax11v 1307 without using ax-11 1008. The hypothesis is even weaker than ax11v 1307, 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 1259.

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 1166	. 2 |- E.z z = y
2		ax11v2.1	. . . . 5 |- (x = z -> (ph -> A.x(x = z -> ph)))
3		equequ2 1177	. . . . . . 7 |- (z = y -> (x = z <-> x = y))
4	3	adantl 397	. . . . . 6 |- ((-. A.x x = y /\ z = y) -> (x = z <-> x = y))
5		dveeq2 1254	. . . . . . . . 9 |- (-. A.x x = y -> (z = y -> A.x z = y))
6	5	imp 357	. . . . . . . 8 |- ((-. A.x x = y /\ z = y) -> A.x z = y)
7		hba1 1044	. . . . . . . . 9 |- (A.x z = y -> A.xA.x z = y)
8	3	imbi1d 624	. . . . . . . . . 10 |- (z = y -> ((x = z -> ph) <-> (x = y -> ph)))
9	8	a4s 1025	. . . . . . . . 9 |- (A.x z = y -> ((x = z -> ph) <-> (x = y -> ph)))
10	7, 9	albid 1145	. . . . . . . 8 |- (A.x z = y -> (A.x(x = z -> ph) <-> A.x(x = y -> ph)))
11	6, 10	syl 10	. . . . . . 7 |- ((-. A.x x = y /\ z = y) -> (A.x(x = z -> ph) <-> A.x(x = y -> ph)))
12	11	imbi2d 623	. . . . . 6 |- ((-. A.x x = y /\ z = y) -> ((ph -> A.x(x = z -> ph)) <-> (ph -> A.x(x = y -> ph))))
13	4, 12	imbi12d 637	. . . . 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 200	. . . 4 |- ((-. A.x x = y /\ z = y) -> (x = y -> (ph -> A.x(x = y -> ph))))
15	14	ex 380	. . 3 |- (-. A.x x = y -> (z = y -> (x = y -> (ph -> A.x(x = y -> ph)))))
16	15	19.23adv 1256	. 2 |- (-. A.x x = y -> (E.z z = y -> (x = y -> (ph -> A.x(x = y -> ph)))))
17	1, 16	mpi 44	1 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995   = wceq 997  E.wex 1021

This theorem is referenced by:  ax11a2 1258

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022

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