Theorem a12lem1 1418

Description: Proof of first hypothesis of a12study 1420.

Assertion

Ref	Expression
a12lem1	|- (-. A.z z = y -> (A.z(z = x -> z = y) -> x = y))

Proof of Theorem a12lem1

Step	Hyp	Ref	Expression
1		equequ1 1176	. . . . . . 7 |- (z = x -> (z = x <-> x = x))
2		equequ1 1176	. . . . . . 7 |- (z = x -> (z = y <-> x = y))
3	1, 2	imbi12d 637	. . . . . 6 |- (z = x -> ((z = x -> z = y) <-> (x = x -> x = y)))
4	3	a4s 1025	. . . . 5 |- (A.z z = x -> ((z = x -> z = y) <-> (x = x -> x = y)))
5	4	dral2 1197	. . . 4 |- (A.z z = x -> (A.z(z = x -> z = y) <-> A.z(x = x -> x = y)))
6		equid 1167	. . . . . . 7 |- x = x
7	6	a1bi 204	. . . . . 6 |- (x = y <-> (x = x -> x = y))
8	7	biimpri 159	. . . . 5 |- ((x = x -> x = y) -> x = y)
9	8	a4s 1025	. . . 4 |- (A.z(x = x -> x = y) -> x = y)
10	5, 9	syl6bi 221	. . 3 |- (A.z z = x -> (A.z(z = x -> z = y) -> x = y))
11	10	a1d 12	. 2 |- (A.z z = x -> (-. A.z z = y -> (A.z(z = x -> z = y) -> x = y)))
12		hbn1 1056	. . . . . . 7 |- (-. A.z z = x -> A.z -. A.z z = x)
13		hbn1 1056	. . . . . . 7 |- (-. A.z z = y -> A.z -. A.z z = y)
14	12, 13	hban 1050	. . . . . 6 |- ((-. A.z z = x /\ -. A.z z = y) -> A.z(-. A.z z = x /\ -. A.z z = y))
15	6	hbth 1042	. . . . . . . 8 |- (x = x -> A.z x = x)
16	15	a1i 8	. . . . . . 7 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = x -> A.z x = x))
17		ax-12 1009	. . . . . . . 8 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
18	17	imp 357	. . . . . . 7 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
19	14, 16, 18	hbimd 1151	. . . . . 6 |- ((-. A.z z = x /\ -. A.z z = y) -> ((x = x -> x = y) -> A.z(x = x -> x = y)))
20	14, 19	19.21ai 1039	. . . . 5 |- ((-. A.z z = x /\ -. A.z z = y) -> A.z((x = x -> x = y) -> A.z(x = x -> x = y)))
21		equtr 1173	. . . . . . . 8 |- (z = x -> (x = x -> z = x))
22		ax-8 1005	. . . . . . . 8 |- (z = x -> (z = y -> x = y))
23	21, 22	imim12d 29	. . . . . . 7 |- (z = x -> ((z = x -> z = y) -> (x = x -> x = y)))
24	23	ax-gen 1004	. . . . . 6 |- A.z(z = x -> ((z = x -> z = y) -> (x = x -> x = y)))
25		19.26 1108	. . . . . . 7 |- (A.z(((x = x -> x = y) -> A.z(x = x -> x = y)) /\ (z = x -> ((z = x -> z = y) -> (x = x -> x = y)))) <-> (A.z((x = x -> x = y) -> A.z(x = x -> x = y)) /\ A.z(z = x -> ((z = x -> z = y) -> (x = x -> x = y)))))
26		a4imt 1200	. . . . . . 7 |- (A.z(((x = x -> x = y) -> A.z(x = x -> x = y)) /\ (z = x -> ((z = x -> z = y) -> (x = x -> x = y)))) -> (A.z(z = x -> z = y) -> (x = x -> x = y)))
27	25, 26	sylbir 208	. . . . . 6 |- ((A.z((x = x -> x = y) -> A.z(x = x -> x = y)) /\ A.z(z = x -> ((z = x -> z = y) -> (x = x -> x = y)))) -> (A.z(z = x -> z = y) -> (x = x -> x = y)))
28	24, 27	mpan2 708	. . . . 5 |- (A.z((x = x -> x = y) -> A.z(x = x -> x = y)) -> (A.z(z = x -> z = y) -> (x = x -> x = y)))
29	20, 28	syl 10	. . . 4 |- ((-. A.z z = x /\ -. A.z z = y) -> (A.z(z = x -> z = y) -> (x = x -> x = y)))
30	6, 29	mpii 45	. . 3 |- ((-. A.z z = x /\ -. A.z z = y) -> (A.z(z = x -> z = y) -> x = y))
31	30	ex 380	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (A.z(z = x -> z = y) -> x = y)))
32	11, 31	pm2.61i 132	1 |- (-. A.z z = y -> (A.z(z = x -> z = y) -> x = y))

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

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-10 1007  ax-12 1009  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

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