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Theorem brtxpsd3 32379
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 328 . . 3 ((𝑥𝐵𝑥𝑋) ↔ (𝑥𝐵𝑥𝑆𝐴))
32albii 1914 . 2 (∀𝑥(𝑥𝐵𝑥𝑋) ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
4 dfcleq 2759 . 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 32378 . 2 (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
103, 4, 93bitr4ri 295 1 (𝐴𝑅𝐵𝐵 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1650   = wceq 1652  wcel 2155  Vcvv 3350  cdif 3729  csymdif 4004   class class class wbr 4809   E cep 5189  ran crn 5278  ctxp 32313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-symdif 4005  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-eprel 5190  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367  df-txp 32337
This theorem is referenced by:  brbigcup  32381  brsingle  32400  brimage  32409  brcart  32415  brapply  32421  brcup  32422  brcap  32423
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