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Theorem brtxpsd3 33254
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 339 . . 3 ((𝑥𝐵𝑥𝑋) ↔ (𝑥𝐵𝑥𝑆𝐴))
32albii 1811 . 2 (∀𝑥(𝑥𝐵𝑥𝑋) ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
4 dfcleq 2812 . 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 33253 . 2 (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
103, 4, 93bitr4ri 305 1 (𝐴𝑅𝐵𝐵 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1526   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  csymdif 4215   class class class wbr 5057   E cep 5457  ran crn 5549  ctxp 33188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-symdif 4216  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679  df-txp 33212
This theorem is referenced by:  brbigcup  33256  brsingle  33275  brimage  33284  brcart  33290  brapply  33296  brcup  33297  brcap  33298
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