Theorem tz6.12i 5891

Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
tz6.12i

Proof of Theorem tz6.12i

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 5881	. . . . 5
2		neeq1 2738	. . . . . . . 8
3		tz6.12-2 5862	. . . . . . . . . . 11
4	3	necon1ai 2688	. . . . . . . . . 10
5		tz6.12c 5890	. . . . . . . . . 10
6	4, 5	syl 16	. . . . . . . . 9
7	6	biimpcd 224	. . . . . . . 8
8	2, 7	sylbird 235	. . . . . . 7
9	8	eqcoms 2469	. . . . . 6
10		neeq1 2738	. . . . . 6
11		breq2 4456	. . . . . 6
12	9, 10, 11	3imtr3d 267	. . . . 5
13	1, 12	vtocle 3183	. . . 4
14	13	a1i 11	. . 3
15		neeq1 2738	. . 3
16		breq2 4456	. . 3
17	14, 15, 16	3imtr3d 267	. 2
18	17	com12 31	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  E!weu 2282  =/=wne 2652  c0 3784   class class class wbr 4452  `cfv 5593

This theorem is referenced by:  fvbr0  5892  fvclss  6154  dcomex  8848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-nul 4581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-iota 5556  df-fv 5601