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Theorem imaeqsalvOLD 7382
Description: Obsolete version of ralima 7255 as of 14-Aug-2025. Duplicate version of ralima 7255. (Contributed by Scott Fenton, 27-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
imaeqsexvOLD.1 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
Assertion
Ref Expression
imaeqsalvOLD ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)

Proof of Theorem imaeqsalvOLD
StepHypRef Expression
1 imaeqsexvOLD.1 . . . . 5 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
21notbid 318 . . . 4 (𝑥 = (𝐹𝑦) → (¬ 𝜑 ↔ ¬ 𝜓))
32imaeqsexvOLD 7381 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵) ¬ 𝜑 ↔ ∃𝑦𝐵 ¬ 𝜓))
43notbid 318 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (¬ ∃𝑥 ∈ (𝐹𝐵) ¬ 𝜑 ↔ ¬ ∃𝑦𝐵 ¬ 𝜓))
5 dfral2 3098 . 2 (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ¬ ∃𝑥 ∈ (𝐹𝐵) ¬ 𝜑)
6 dfral2 3098 . 2 (∀𝑦𝐵 𝜓 ↔ ¬ ∃𝑦𝐵 ¬ 𝜓)
74, 5, 63bitr4g 314 1 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wral 3060  wrex 3069  wss 3950  cima 5686   Fn wfn 6554  cfv 6559
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 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-fv 6567
This theorem is referenced by: (None)
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