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Theorem brtxpsd3 35897
Description: A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
Hypotheses
Ref Expression
brtxpsd2.1 𝐴 ∈ V
brtxpsd2.2 𝐵 ∈ V
brtxpsd2.3 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))
brtxpsd2.4 𝐴𝐶𝐵
brtxpsd3.5 (𝑥𝑋𝑥𝑆𝐴)
Assertion
Ref Expression
brtxpsd3 (𝐴𝑅𝐵𝐵 = 𝑋)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑆   𝑥,𝑋
Allowed substitution hints:   𝐶(𝑥)   𝑅(𝑥)

Proof of Theorem brtxpsd3
StepHypRef Expression
1 brtxpsd3.5 . . . 4 (𝑥𝑋𝑥𝑆𝐴)
21bibi2i 337 . . 3 ((𝑥𝐵𝑥𝑋) ↔ (𝑥𝐵𝑥𝑆𝐴))
32albii 1819 . 2 (∀𝑥(𝑥𝐵𝑥𝑋) ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
4 dfcleq 2730 . 2 (𝐵 = 𝑋 ↔ ∀𝑥(𝑥𝐵𝑥𝑋))
5 brtxpsd2.1 . . 3 𝐴 ∈ V
6 brtxpsd2.2 . . 3 𝐵 ∈ V
7 brtxpsd2.3 . . 3 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))
8 brtxpsd2.4 . . 3 𝐴𝐶𝐵
95, 6, 7, 8brtxpsd2 35896 . 2 (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
103, 4, 93bitr4ri 304 1 (𝐴𝑅𝐵𝐵 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  csymdif 4252   class class class wbr 5143   E cep 5583  ran crn 5686  ctxp 35831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-symdif 4253  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855
This theorem is referenced by:  brbigcup  35899  brsingle  35918  brimage  35927  brcart  35933  brapply  35939  brcup  35940  brcap  35941
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