Theorem a12studyALT 1421

Description: Alternate proof of a12study 1420, also without using ax-12 1009.

Hypotheses

Ref	Expression
a12study.1	|- (-. A.z z = y -> (A.z(z = x -> z = y) -> x = y))
a12study.2	|- (A.z(z = x -> -. z = y) -> -. x = y)

Assertion

Ref	Expression
a12studyALT	|- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))

Proof of Theorem a12studyALT

Step	Hyp	Ref	Expression
1		hbn1 1056	. . . . 5 |- (-. A.z z = x -> A.z -. A.z z = x)
2		hbn1 1056	. . . . 5 |- (-. A.z z = y -> A.z -. A.z z = y)
3	1, 2	hban 1050	. . . 4 |- ((-. A.z z = x /\ -. A.z z = y) -> A.z(-. A.z z = x /\ -. A.z z = y))
4		a12study.1	. . . . . 6 |- (-. A.z z = y -> (A.z(z = x -> z = y) -> x = y))
5	4	con3d 99	. . . . 5 |- (-. A.z z = y -> (-. x = y -> -. A.z(z = x -> z = y)))
6		hba1 1044	. . . . . . 7 |- (A.z -. x = y -> A.zA.z -. x = y)
7		ax-11o 1260	. . . . . . . . . . 11 |- (-. A.z z = x -> (z = x -> (z = y -> A.z(z = x -> z = y))))
8	7	ax11indn 1408	. . . . . . . . . 10 |- (-. A.z z = x -> (z = x -> (-. z = y -> A.z(z = x -> -. z = y))))
9		a12study.2	. . . . . . . . . . 11 |- (A.z(z = x -> -. z = y) -> -. x = y)
10	9	a5i 1030	. . . . . . . . . 10 |- (A.z(z = x -> -. z = y) -> A.z -. x = y)
11	8, 10	syl8 24	. . . . . . . . 9 |- (-. A.z z = x -> (z = x -> (-. z = y -> A.z -. x = y)))
12	11	imp3a 368	. . . . . . . 8 |- (-. A.z z = x -> ((z = x /\ -. z = y) -> A.z -. x = y))
13		annim 245	. . . . . . . 8 |- ((z = x /\ -. z = y) <-> -. (z = x -> z = y))
14	12, 13	syl5ibr 214	. . . . . . 7 |- (-. A.z z = x -> (-. (z = x -> z = y) -> A.z -. x = y))
15	1, 6, 14	19.23ad 1107	. . . . . 6 |- (-. A.z z = x -> (E.z -. (z = x -> z = y) -> A.z -. x = y))
16		exnal 1079	. . . . . 6 |- (E.z -. (z = x -> z = y) <-> -. A.z(z = x -> z = y))
17	15, 16	syl5ibr 214	. . . . 5 |- (-. A.z z = x -> (-. A.z(z = x -> z = y) -> A.z -. x = y))
18	5, 17	sylan9r 480	. . . 4 |- ((-. A.z z = x /\ -. A.z z = y) -> (-. x = y -> A.z -. x = y))
19	3, 18	hbnd 1150	. . 3 |- ((-. A.z z = x /\ -. A.z z = y) -> (-. -. x = y -> A.z -. -. x = y))
20		notnot 168	. . 3 |- (x = y <-> -. -. x = y)
21	20	albii 1040	. . 3 |- (A.z x = y <-> A.z -. -. x = y)
22	19, 20, 21	3imtr4g 564	. 2 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
23	22	ex 380	1 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))

Syntax hints:  -. wn 2   -> wi 3   /\ 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-gen 1004  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-11o 1260

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

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