Theorem ax4 1013

Description: Theorem showing that ax-4 1014 can be derived from ax-5 1001, ax-gen 1004, ax-8 1005, ax-9 1006, ax-11 1008, and ax-17 1012 and is therefore redundant. Lemma 21 of [Monk2] p. 114.

This theorem should not be referenced in any proof. Instead, we will use ax-4 1014 below so that uses of ax-4 1014 can be more easily identified. In particular, this will more cleanly separate out the theorems of "pure" predicate calculus that don't involve equality or distinct variables. A beginner can accept ax-4 1014 a priori, so that the proof of this theorem (ax4 1013), which involves equality as well as the distinct the distinct variable requirements of ax-17 1012, can be put off until later.

Note: All predicate calculus axioms introduced from this point forward are redundant. Immediately before their introduction, we prove them from earlier axioms to demonstrate their redundancy. Specifically, redundant axioms ax-4 1014, ax-5o 1016, ax-6o 1019, ax-9o 1164, ax-10o 1182, ax-11o 1260, ax-15 1402, and ax-16 1252 are proved by theorems ax4 1013, ax5o 1015, ax6o 1018, ax9o 1163, ax10o 1181, ax11o 1259, ax15 1401, and ax16 1251.

Assertion

Ref	Expression
ax4	|- (A.xph -> ph)

Proof of Theorem ax4

Step	Hyp	Ref	Expression
1		ax-9 1006	. 2 |- -. A.y -. y = x
2		ax-17 1012	. . 3 |- (-. (A.xph -> ph) -> A.y -. (A.xph -> ph))
3		ax-9 1006	. . . . . . . . . . . . . 14 |- -. A.x -. x = y
4		ax-17 1012	. . . . . . . . . . . . . . 15 |- (-. y = y -> A.x -. y = y)
5		ax-8 1005	. . . . . . . . . . . . . . . . . . 19 |- (x = y -> (x = y -> y = y))
6	5	pm2.43i 64	. . . . . . . . . . . . . . . . . 18 |- (x = y -> y = y)
7	6	con3i 104	. . . . . . . . . . . . . . . . 17 |- (-. y = y -> -. x = y)
8	7	ax-gen 1004	. . . . . . . . . . . . . . . 16 |- A.x(-. y = y -> -. x = y)
9		ax-5 1001	. . . . . . . . . . . . . . . 16 |- (A.x(-. y = y -> -. x = y) -> (A.x -. y = y -> A.x -. x = y))
10	8, 9	ax-mp 7	. . . . . . . . . . . . . . 15 |- (A.x -. y = y -> A.x -. x = y)
11	4, 10	syl 10	. . . . . . . . . . . . . 14 |- (-. y = y -> A.x -. x = y)
12	3, 11	mt3 118	. . . . . . . . . . . . 13 |- y = y
13		ax-8 1005	. . . . . . . . . . . . 13 |- (y = x -> (y = y -> x = y))
14	12, 13	mpi 44	. . . . . . . . . . . 12 |- (y = x -> x = y)
15		ax-11 1008	. . . . . . . . . . . 12 |- (x = y -> (A.y -. ph -> A.x(x = y -> -. ph)))
16	14, 15	syl 10	. . . . . . . . . . 11 |- (y = x -> (A.y -. ph -> A.x(x = y -> -. ph)))
17		ax-17 1012	. . . . . . . . . . 11 |- (-. ph -> A.y -. ph)
18	16, 17	syl5 21	. . . . . . . . . 10 |- (y = x -> (-. ph -> A.x(x = y -> -. ph)))
19		con2 94	. . . . . . . . . . . 12 |- ((x = y -> -. ph) -> (ph -> -. x = y))
20	19	ax-gen 1004	. . . . . . . . . . 11 |- A.x((x = y -> -. ph) -> (ph -> -. x = y))
21		ax-5 1001	. . . . . . . . . . 11 |- (A.x((x = y -> -. ph) -> (ph -> -. x = y)) -> (A.x(x = y -> -. ph) -> A.x(ph -> -. x = y)))
22	20, 21	ax-mp 7	. . . . . . . . . 10 |- (A.x(x = y -> -. ph) -> A.x(ph -> -. x = y))
23	18, 22	syl6 22	. . . . . . . . 9 |- (y = x -> (-. ph -> A.x(ph -> -. x = y)))
24		ax-5 1001	. . . . . . . . 9 |- (A.x(ph -> -. x = y) -> (A.xph -> A.x -. x = y))
25	23, 24	syl6 22	. . . . . . . 8 |- (y = x -> (-. ph -> (A.xph -> A.x -. x = y)))
26		con3 98	. . . . . . . . 9 |- ((A.xph -> A.x -. x = y) -> (-. A.x -. x = y -> -. A.xph))
27	3, 26	mpi 44	. . . . . . . 8 |- ((A.xph -> A.x -. x = y) -> -. A.xph)
28	25, 27	syl6 22	. . . . . . 7 |- (y = x -> (-. ph -> -. A.xph))
29	28	a3d 78	. . . . . 6 |- (y = x -> (A.xph -> ph))
30	29	con3i 104	. . . . 5 |- (-. (A.xph -> ph) -> -. y = x)
31	30	ax-gen 1004	. . . 4 |- A.y(-. (A.xph -> ph) -> -. y = x)
32		ax-5 1001	. . . 4 |- (A.y(-. (A.xph -> ph) -> -. y = x) -> (A.y -. (A.xph -> ph) -> A.y -. y = x))
33	31, 32	ax-mp 7	. . 3 |- (A.y -. (A.xph -> ph) -> A.y -. y = x)
34	2, 33	syl 10	. 2 |- (-. (A.xph -> ph) -> A.y -. y = x)
35	1, 34	mt3 118	1 |- (A.xph -> ph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 995

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1001  ax-gen 1004  ax-8 1005  ax-9 1006  ax-11 1008  ax-17 1012

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