Theorem ax11inda2ALT 1760

Description: A proof of ax11inda2 1761 that is slightly more direct.

Hypothesis

Ref	Expression
ax11inda2.1	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Assertion

Ref	Expression
ax11inda2ALT	|- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

Distinct variable group:   y,z

Proof of Theorem ax11inda2ALT

Step	Hyp	Ref	Expression
1		ax-1 4	. . . . . . . 8 |- (A.xph -> (x = y -> A.xph))
2	1	a5i 1335	. . . . . . 7 |- (A.xph -> A.x(x = y -> A.xph))
3	2	a1i 8	. . . . . 6 |- (A.z z = x -> (A.xph -> A.x(x = y -> A.xph)))
4		biidd 188	. . . . . . 7 |- (A.z z = x -> (ph <-> ph))
5	4	dral1 1515	. . . . . 6 |- (A.z z = x -> (A.zph <-> A.xph))
6	5	imbi2d 674	. . . . . . 7 |- (A.z z = x -> ((x = y -> A.zph) <-> (x = y -> A.xph)))
7	6	dral2 1516	. . . . . 6 |- (A.z z = x -> (A.x(x = y -> A.zph) <-> A.x(x = y -> A.xph)))
8	3, 5, 7	3imtr4d 602	. . . . 5 |- (A.z z = x -> (A.zph -> A.x(x = y -> A.zph)))
9	8	alequcoms 1503	. . . 4 |- (A.x x = z -> (A.zph -> A.x(x = y -> A.zph)))
10	9	a1d 15	. . 3 |- (A.x x = z -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))
11	10	a1d 15	. 2 |- (A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
12		simplr 449	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> -. A.x x = y)
13		dveeq1 1745	. . . . . . . 8 |- (-. A.z z = x -> (x = y -> A.z x = y))
14	13	nalequcoms 1504	. . . . . . 7 |- (-. A.x x = z -> (x = y -> A.z x = y))
15	14	imp 377	. . . . . 6 |- ((-. A.x x = z /\ x = y) -> A.z x = y)
16	15	adantlr 429	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> A.z x = y)
17		hbnae 1507	. . . . . . 7 |- (-. A.x x = y -> A.z -. A.x x = y)
18		hba1 1350	. . . . . . 7 |- (A.z x = y -> A.zA.z x = y)
19	17, 18	hban 1356	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> A.z(-. A.x x = y /\ A.z x = y))
20		ax11inda2.1	. . . . . . . 8 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
21	20	imp 377	. . . . . . 7 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
22		ax-4 1319	. . . . . . 7 |- (A.z x = y -> x = y)
23	21, 22	sylan2 500	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> (ph -> A.x(x = y -> ph)))
24	19, 23	alimd 1343	. . . . 5 |- ((-. A.x x = y /\ A.z x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
25	12, 16, 24	syl11anc 524	. . . 4 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
26		hbnae 1507	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
27		hbnae 1507	. . . . . . . . 9 |- (-. A.x x = z -> A.z -. A.x x = z)
28	27, 14	19.21ai 1345	. . . . . . . 8 |- (-. A.x x = z -> A.z(x = y -> A.z x = y))
29		19.21t 1473	. . . . . . . 8 |- (A.z(x = y -> A.z x = y) -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
30	28, 29	syl 12	. . . . . . 7 |- (-. A.x x = z -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
31	26, 30	albid 1459	. . . . . 6 |- (-. A.x x = z -> (A.xA.z(x = y -> ph) <-> A.x(x = y -> A.zph)))
32		ax-7 1304	. . . . . 6 |- (A.zA.x(x = y -> ph) -> A.xA.z(x = y -> ph))
33	31, 32	syl5bi 225	. . . . 5 |- (-. A.x x = z -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
34	33	ad2antrr 440	. . . 4 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
35	25, 34	syld 30	. . 3 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.x(x = y -> A.zph)))
36	35	exp31 407	. 2 |- (-. A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
37	11, 36	pm2.61i 140	1 |- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500

This theorem depends on definitions:  df-bi 164  df-an 242

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