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Theorem brtxpsd2 33466
Description: Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
Hypotheses
Ref Expression
brtxpsd2.1 𝐴 ∈ V
brtxpsd2.2 𝐵 ∈ V
brtxpsd2.3 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))
brtxpsd2.4 𝐴𝐶𝐵
Assertion
Ref Expression
brtxpsd2 (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑆
Allowed substitution hints:   𝐶(𝑥)   𝑅(𝑥)

Proof of Theorem brtxpsd2
StepHypRef Expression
1 brtxpsd2.4 . . 3 𝐴𝐶𝐵
2 brtxpsd2.3 . . . . 5 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))
32breqi 5036 . . . 4 (𝐴𝑅𝐵𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵)
4 brdif 5083 . . . 4 (𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵 ↔ (𝐴𝐶𝐵 ∧ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵))
53, 4bitri 278 . . 3 (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐵 ∧ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵))
61, 5mpbiran 708 . 2 (𝐴𝑅𝐵 ↔ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵)
7 brtxpsd2.1 . . 3 𝐴 ∈ V
8 brtxpsd2.2 . . 3 𝐵 ∈ V
97, 8brtxpsd 33465 . 2 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
106, 9bitri 278 1 (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥𝐵𝑥𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2111  Vcvv 3441  cdif 3878  csymdif 4168   class class class wbr 5030   E cep 5429  ran crn 5520  ctxp 33401
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-symdif 4169  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-eprel 5430  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-1st 7671  df-2nd 7672  df-txp 33425
This theorem is referenced by:  brtxpsd3  33467
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