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Theorem brwitnlem 6743
 Description: Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
brwitnlem.r
brwitnlem.o
Assertion
Ref Expression
brwitnlem

Proof of Theorem brwitnlem
StepHypRef Expression
1 fvex 5734 . . . . 5
2 dif1o 6736 . . . . 5
31, 2mpbiran 885 . . . 4
43anbi2i 676 . . 3
5 brwitnlem.o . . . 4
6 elpreima 5842 . . . 4
75, 6ax-mp 8 . . 3
8 ndmfv 5747 . . . . . 6
98necon1ai 2640 . . . . 5
10 fndm 5536 . . . . . 6
115, 10ax-mp 8 . . . . 5
129, 11syl6eleq 2525 . . . 4
1312pm4.71ri 615 . . 3
144, 7, 133bitr4i 269 . 2
15 brwitnlem.r . . . 4
1615breqi 4210 . . 3
17 df-br 4205 . . 3
1816, 17bitri 241 . 2
19 df-ov 6076 . . 3
2019neeq1i 2608 . 2
2114, 18, 203bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725   wne 2598  cvv 2948   cdif 3309  c0 3620  cop 3809   class class class wbr 4204  ccnv 4869   cdm 4870  cima 4873   wfn 5441  cfv 5446  (class class class)co 6073  c1o 6709 This theorem is referenced by:  brgic  15048  brlmic  16132  hmph  17800 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ov 6076  df-1o 6716
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