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Theorem brpprod3a 25733
 Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brpprod3.1
brpprod3.2
brpprod3.3
Assertion
Ref Expression
brpprod3a pprod
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,

Proof of Theorem brpprod3a
StepHypRef Expression
1 pprodss4v 25731 . . . . . . 7 pprod
21brel 4928 . . . . . 6 pprod
32simprd 451 . . . . 5 pprod
4 elvv 4938 . . . . 5
53, 4sylib 190 . . . 4 pprod
65pm4.71ri 616 . . 3 pprod pprod
7 19.41vv 1926 . . 3 pprod pprod
86, 7bitr4i 245 . 2 pprod pprod
9 breq2 4218 . . . 4 pprod pprod
109pm5.32i 620 . . 3 pprod pprod
11102exbii 1594 . 2 pprod pprod
12 brpprod3.1 . . . . . 6
13 brpprod3.2 . . . . . 6
14 vex 2961 . . . . . 6
15 vex 2961 . . . . . 6
1612, 13, 14, 15brpprod 25732 . . . . 5 pprod
1716anbi2i 677 . . . 4 pprod
18 3anass 941 . . . 4
1917, 18bitr4i 245 . . 3 pprod
20192exbii 1594 . 2 pprod
218, 11, 203bitri 264 1 pprod
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726  cvv 2958  cop 3819   class class class wbr 4214   cxp 4878  pprodcpprod 25677 This theorem is referenced by:  brpprod3b  25734  brapply  25785  dfrdg4  25797 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-txp 25700  df-pprod 25701
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