Theorem ax4 1156

Description: Theorem showing that ax-4 1157 can be derived from ax-5 1140, ax-gen 1143, ax-8 1144, ax-9 1145, ax-11 1147, and ax-17 1155 and is therefore redundant in a system including the latter axioms. Lemma 21 of [Monk2] p. 114.

This theorem should not be referenced in any proof. Instead, we will use ax-4 1157 below so that explicit uses of ax-4 1157 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 may wish to accept ax-4 1157 a priori, so that the proof of this theorem (ax4 1156), which involves equality as well as the distinct variable requirements of ax-17 1155, can be put off until those axioms are studied.

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 1157, ax-5o 1159, ax-6o 1162, ax-9o 1319, ax-10o 1338, ax-11o 1426, ax-15 1589, and ax-16 1418 are proved by theorems ax4 1156, ax5o 1158, ax6o 1161, ax9o 1318, ax10o 1337, ax11o 1425, ax15 1588, and ax16 1417. We never reference those theorems directly - instead, we use the axiom version that immediately follows it - in order to better isolate the uses of the redundant axioms for easier study of subsystems containing them.

(The proof was shortened by Scott Fenton, 24-Jan-2011.)

Assertion

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

Proof of Theorem ax4

Step	Hyp	Ref	Expression
1		ax-9 1145	. 2 |- -. A.y -. y = x
2		ax-17 1155	. . 3 |- (-. (A.xph -> ph) -> A.y -. (A.xph -> ph))
3		ax-9 1145	. . . . . . . . . . . 12 |- -. A.x -. x = y
4		ax-17 1155	. . . . . . . . . . . . 13 |- (-. y = y -> A.x -. y = y)
5		ax-8 1144	. . . . . . . . . . . . . . . . 17 |- (x = y -> (x = y -> y = y))
6	5	pm2.43i 78	. . . . . . . . . . . . . . . 16 |- (x = y -> y = y)
7	6	con3i 113	. . . . . . . . . . . . . . 15 |- (-. y = y -> -. x = y)
8	7	ax-gen 1143	. . . . . . . . . . . . . 14 |- A.x(-. y = y -> -. x = y)
9		ax-5 1140	. . . . . . . . . . . . . 14 |- (A.x(-. y = y -> -. x = y) -> (A.x -. y = y -> A.x -. x = y))
10	8, 9	ax-mp 7	. . . . . . . . . . . . 13 |- (A.x -. y = y -> A.x -. x = y)
11	4, 10	syl 12	. . . . . . . . . . . 12 |- (-. y = y -> A.x -. x = y)
12	3, 11	mt3 126	. . . . . . . . . . 11 |- y = y
13		ax-8 1144	. . . . . . . . . . 11 |- (y = x -> (y = y -> x = y))
14	12, 13	mpi 55	. . . . . . . . . 10 |- (y = x -> x = y)
15		ax-11 1147	. . . . . . . . . 10 |- (x = y -> (A.y -. ph -> A.x(x = y -> -. ph)))
16	14, 15	syl 12	. . . . . . . . 9 |- (y = x -> (A.y -. ph -> A.x(x = y -> -. ph)))
17		ax-17 1155	. . . . . . . . 9 |- (-. ph -> A.y -. ph)
18	16, 17	syl5 20	. . . . . . . 8 |- (y = x -> (-. ph -> A.x(x = y -> -. ph)))
19		con2 105	. . . . . . . . . . . 12 |- ((x = y -> -. ph) -> (ph -> -. x = y))
20	19	ax-gen 1143	. . . . . . . . . . 11 |- A.x((x = y -> -. ph) -> (ph -> -. x = y))
21		ax-5 1140	. . . . . . . . . . 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		ax-5 1140	. . . . . . . . . 10 |- (A.x(ph -> -. x = y) -> (A.xph -> A.x -. x = y))
24	22, 23	syl 12	. . . . . . . . 9 |- (A.x(x = y -> -. ph) -> (A.xph -> A.x -. x = y))
25	3, 24	mtoi 121	. . . . . . . 8 |- (A.x(x = y -> -. ph) -> -. A.xph)
26	18, 25	syl6 25	. . . . . . 7 |- (y = x -> (-. ph -> -. A.xph))
27	26	con4d 90	. . . . . 6 |- (y = x -> (A.xph -> ph))
28	27	con3i 113	. . . . 5 |- (-. (A.xph -> ph) -> -. y = x)
29	28	ax-gen 1143	. . . 4 |- A.y(-. (A.xph -> ph) -> -. y = x)
30		ax-5 1140	. . . 4 |- (A.y(-. (A.xph -> ph) -> -. y = x) -> (A.y -. (A.xph -> ph) -> A.y -. y = x))
31	29, 30	ax-mp 7	. . 3 |- (A.y -. (A.xph -> ph) -> A.y -. y = x)
32	2, 31	syl 12	. 2 |- (-. (A.xph -> ph) -> A.y -. y = x)
33	1, 32	mt3 126	1 |- (A.xph -> ph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 1134

This theorem is referenced by:  abidhb 2256  fundmpss 13629  iotain 16063

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1140  ax-gen 1143  ax-8 1144  ax-9 1145  ax-11 1147  ax-17 1155

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