Theorem inecmo 34120

Description: Lemma for ~? dfeldisj5 (via inecmo2 34121), ~? dfdisjs5 , ~? dfdisjALTV5 , ~? eldisjs5 (via inecmo3 34126, ~? cosscnvssid5 ), ~? dffunsALTV5 (via ineccnvmo 34122, ineccnvmo2 34125), and ~? dffunALTV5 (via ~? cossssid5 ). (Contributed by Peter Mazsa, 29-May-2018.)

Hypothesis

Ref	Expression
inecmo.1	|- ( x = y -> B = C )

Assertion

Ref	Expression
inecmo	|- ( Rel R -> ( A. x e. A A. y e. A ( x = y \/ ( [ B ] R i^i [ C ] R ) = (/) ) <-> A. z E* x e. A B R z ) )

Distinct variable groups:   x, A, y, z   y, B, z   x, C, z   x, R, y, z

Allowed substitution hints:   B( x)   C( y)

Proof of Theorem inecmo

Step	Hyp	Ref	Expression
1		relelec 7787	. . . . . . 7 |- ( Rel R -> ( z e. [ B ] R <-> B R z ) )
2		relelec 7787	. . . . . . 7 |- ( Rel R -> ( z e. [ C ] R <-> C R z ) )
3	1, 2	anbi12d 747	. . . . . 6 |- ( Rel R -> ( ( z e. [ B ] R /\ z e. [ C ] R ) <-> ( B R z /\ C R z ) ) )
4	3	imbi1d 331	. . . . 5 |- ( Rel R -> ( ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) <-> ( ( B R z /\ C R z ) -> x = y ) ) )
5	4	2ralbidv 2989	. . . 4 |- ( Rel R -> ( A. x e. A A. y e. A ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) <-> A. x e. A A. y e. A ( ( B R z /\ C R z ) -> x = y ) ) )
6		inecmo.1	. . . . . 6 |- ( x = y -> B = C )
7	6	breq1d 4663	. . . . 5 |- ( x = y -> ( B R z <-> C R z ) )
8	7	rmo4 3399	. . . 4 |- ( E* x e. A B R z <-> A. x e. A A. y e. A ( ( B R z /\ C R z ) -> x = y ) )
9	5, 8	syl6rbbr 279	. . 3 |- ( Rel R -> ( E* x e. A B R z <-> A. x e. A A. y e. A ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) ) )
10	9	albidv 1849	. 2 |- ( Rel R -> ( A. z E* x e. A B R z <-> A. z A. x e. A A. y e. A ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) ) )
11		ineleq 34119	. . 3 |- ( A. x e. A A. y e. A ( x = y \/ ( [ B ] R i^i [ C ] R ) = (/) ) <-> A. x e. A A. z A. y e. A ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) )
12		ralcom4 3224	. . 3 |- ( A. x e. A A. z A. y e. A ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) <-> A. z A. x e. A A. y e. A ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) )
13	11, 12	bitri 264	. 2 |- ( A. x e. A A. y e. A ( x = y \/ ( [ B ] R i^i [ C ] R ) = (/) ) <-> A. z A. x e. A A. y e. A ( ( z e. [ B ] R /\ z e. [ C ] R ) -> x = y ) )
14	10, 13	syl6rbbr 279	1 |- ( Rel R -> ( A. x e. A A. y e. A ( x = y \/ ( [ B ] R i^i [ C ] R ) = (/) ) <-> A. z E* x e. A B R z ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   E*wrmo 2915   i^i cin 3573   (/)c0 3915   class class class wbr 4653   Rel wrel 5119   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  inecmo2  34121  ineccnvmo  34122