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Theorem brtxpsd3 33364
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 341 . . 3 ((𝑥𝐵𝑥𝑋) ↔ (𝑥𝐵𝑥𝑆𝐴))
32albii 1821 . 2 (∀𝑥(𝑥𝐵𝑥𝑋) ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
4 dfcleq 2815 . 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 33363 . 2 (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
103, 4, 93bitr4ri 307 1 (𝐴𝑅𝐵𝐵 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536   = wceq 1538  wcel 2115  Vcvv 3471  cdif 3907  csymdif 4193   class class class wbr 5039   E cep 5437  ran crn 5529  ctxp 33298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-symdif 4194  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-eprel 5438  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-1st 7664  df-2nd 7665  df-txp 33322
This theorem is referenced by:  brbigcup  33366  brsingle  33385  brimage  33394  brcart  33400  brapply  33406  brcup  33407  brcap  33408
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