Theorem ax11inda 1413

Description: Induction step for constructing a substitution instance of ax-11o 1260 without using ax-11o 1260. Quantification case. (When z and y are distinct, ax11inda2 1412 may be used instead to avoid the dummy variable w in the proof.)

Hypothesis

Ref	Expression
ax11inda.1	|- (-. A.x x = w -> (x = w -> (ph -> A.x(x = w -> ph))))

Assertion

Ref	Expression
ax11inda	|- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

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

Proof of Theorem ax11inda

Step	Hyp	Ref	Expression
1		a9e 1166	. . 3 |- E.w w = y
2		ax11inda.1	. . . . . . 7 |- (-. A.x x = w -> (x = w -> (ph -> A.x(x = w -> ph))))
3	2	ax11inda2 1412	. . . . . 6 |- (-. A.x x = w -> (x = w -> (A.zph -> A.x(x = w -> A.zph))))
4		dveeq2 1254	. . . . . . . . 9 |- (-. A.x x = y -> (w = y -> A.x w = y))
5	4	imp 357	. . . . . . . 8 |- ((-. A.x x = y /\ w = y) -> A.x w = y)
6		hba1 1044	. . . . . . . . . 10 |- (A.x w = y -> A.xA.x w = y)
7		equequ2 1177	. . . . . . . . . . 11 |- (w = y -> (x = w <-> x = y))
8	7	a4s 1025	. . . . . . . . . 10 |- (A.x w = y -> (x = w <-> x = y))
9	6, 8	albid 1145	. . . . . . . . 9 |- (A.x w = y -> (A.x x = w <-> A.x x = y))
10	9	notbid 622	. . . . . . . 8 |- (A.x w = y -> (-. A.x x = w <-> -. A.x x = y))
11	5, 10	syl 10	. . . . . . 7 |- ((-. A.x x = y /\ w = y) -> (-. A.x x = w <-> -. A.x x = y))
12	7	adantl 397	. . . . . . . 8 |- ((-. A.x x = y /\ w = y) -> (x = w <-> x = y))
13	8	imbi1d 624	. . . . . . . . . . 11 |- (A.x w = y -> ((x = w -> A.zph) <-> (x = y -> A.zph)))
14	6, 13	albid 1145	. . . . . . . . . 10 |- (A.x w = y -> (A.x(x = w -> A.zph) <-> A.x(x = y -> A.zph)))
15	5, 14	syl 10	. . . . . . . . 9 |- ((-. A.x x = y /\ w = y) -> (A.x(x = w -> A.zph) <-> A.x(x = y -> A.zph)))
16	15	imbi2d 623	. . . . . . . 8 |- ((-. A.x x = y /\ w = y) -> ((A.zph -> A.x(x = w -> A.zph)) <-> (A.zph -> A.x(x = y -> A.zph))))
17	12, 16	imbi12d 637	. . . . . . 7 |- ((-. A.x x = y /\ w = y) -> ((x = w -> (A.zph -> A.x(x = w -> A.zph))) <-> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
18	11, 17	imbi12d 637	. . . . . 6 |- ((-. A.x x = y /\ w = y) -> ((-. A.x x = w -> (x = w -> (A.zph -> A.x(x = w -> A.zph)))) <-> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))))
19	3, 18	mpbii 200	. . . . 5 |- ((-. A.x x = y /\ w = y) -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
20	19	ex 380	. . . 4 |- (-. A.x x = y -> (w = y -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))))
21	20	19.23adv 1256	. . 3 |- (-. A.x x = y -> (E.w w = y -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))))
22	1, 21	mpi 44	. 2 |- (-. A.x x = y -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
23	22	pm2.43i 64	1 |- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

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

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  ax-16 1252

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

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