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Theorem brpprod3b 25732
 Description: Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
Hypotheses
Ref Expression
brpprod3.1
brpprod3.2
brpprod3.3
Assertion
Ref Expression
brpprod3b pprod
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,

Proof of Theorem brpprod3b
StepHypRef Expression
1 pprodcnveq 25728 . . 3 pprod pprod
21breqi 4218 . 2 pprod pprod
3 brpprod3.1 . . . . 5
4 opex 4427 . . . . 5
53, 4brcnv 5055 . . . 4 pprod pprod
6 brpprod3.2 . . . . 5
7 brpprod3.3 . . . . 5
86, 7, 3brpprod3a 25731 . . . 4 pprod
95, 8bitri 241 . . 3 pprod
10 biid 228 . . . . 5
11 vex 2959 . . . . . 6
126, 11brcnv 5055 . . . . 5
13 vex 2959 . . . . . 6
147, 13brcnv 5055 . . . . 5
1510, 12, 143anbi123i 1142 . . . 4
16152exbii 1593 . . 3
179, 16bitri 241 . 2 pprod
182, 17bitri 241 1 pprod
 Colors of variables: wff set class Syntax hints:   wb 177   w3a 936  wex 1550   wceq 1652   wcel 1725  cvv 2956  cop 3817   class class class wbr 4212  ccnv 4877  pprodcpprod 25675 This theorem is referenced by:  brcart  25777 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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-txp 25698  df-pprod 25699
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