Theorem a12study 1420

Description: Rederivation of axiom ax-12 1009 from two shorter formulas, without using ax-12 1009. See a12lem1 1418 and a12lem2 1419 for the proofs of the hypotheses (using ax-12 1009). This is the only known breakdown of ax-12 1009 into shorter formulas. See a12studyALT 1421 for an alternate proof. Note that the proof depends on ax-11o 1260, whose proof ax11o 1259 depends on ax-12 1009, meaning that we would have to replace ax-11 1008 with ax-11o 1260 in an axiomatization that uses the hypotheses in place of ax-12 1009. Whether this can be avoided is an open problem.

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
a12study	|- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))

Proof of Theorem a12study

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		hba1 1044	. . . 4 |- (A.z x = y -> A.zA.z x = y)
5		ax-11o 1260	. . . . . . 7 |- (-. A.z z = x -> (z = x -> (z = y -> A.z(z = x -> z = y))))
6		equid1 1311	. . . . . . . 8 |- x = x
7		ax-8 1005	. . . . . . . 8 |- (x = z -> (x = x -> z = x))
8	6, 7	mpi 44	. . . . . . 7 |- (x = z -> z = x)
9	5, 8	syl5 21	. . . . . 6 |- (-. A.z z = x -> (x = z -> (z = y -> A.z(z = x -> z = y))))
10	9	imp3a 368	. . . . 5 |- (-. A.z z = x -> ((x = z /\ z = y) -> A.z(z = x -> z = y)))
11		hba1 1044	. . . . . 6 |- (A.z(z = x -> z = y) -> A.zA.z(z = x -> z = y))
12		a12study.1	. . . . . 6 |- (-. A.z z = y -> (A.z(z = x -> z = y) -> x = y))
13	2, 11, 12	19.21ad 1100	. . . . 5 |- (-. A.z z = y -> (A.z(z = x -> z = y) -> A.z x = y))
14	10, 13	sylan9 479	. . . 4 |- ((-. A.z z = x /\ -. A.z z = y) -> ((x = z /\ z = y) -> A.z x = y))
15	3, 4, 14	19.23ad 1107	. . 3 |- ((-. A.z z = x /\ -. A.z z = y) -> (E.z(x = z /\ z = y) -> A.z x = y))
16	15	ex 380	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (E.z(x = z /\ z = y) -> A.z x = y)))
17		imnan 249	. . . . . . 7 |- ((x = z -> -. z = y) <-> -. (x = z /\ z = y))
18		equid1 1311	. . . . . . . . 9 |- z = z
19		ax-8 1005	. . . . . . . . 9 |- (z = x -> (z = z -> x = z))
20	18, 19	mpi 44	. . . . . . . 8 |- (z = x -> x = z)
21	20	imim1i 16	. . . . . . 7 |- ((x = z -> -. z = y) -> (z = x -> -. z = y))
22	17, 21	sylbir 208	. . . . . 6 |- (-. (x = z /\ z = y) -> (z = x -> -. z = y))
23	22	19.20i 1033	. . . . 5 |- (A.z -. (x = z /\ z = y) -> A.z(z = x -> -. z = y))
24		a12study.2	. . . . 5 |- (A.z(z = x -> -. z = y) -> -. x = y)
25	23, 24	syl 10	. . . 4 |- (A.z -. (x = z /\ z = y) -> -. x = y)
26	25	con2i 102	. . 3 |- (x = y -> -. A.z -. (x = z /\ z = y))
27		df-ex 1022	. . 3 |- (E.z(x = z /\ z = y) <-> -. A.z -. (x = z /\ z = y))
28	26, 27	sylibr 207	. 2 |- (x = y -> E.z(x = z /\ z = y))
29	16, 28	syl7 23	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-8 1005  ax-9 1006  ax-17 1012  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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