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Theorem imaeqsexvOLD 7381
Description: Obsolete version of rexima 7256 as of 14-Aug-2025. Duplicate version of rexima 7256. (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
imaeqsexvOLD ((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 imaeqsexvOLD
StepHypRef Expression
1 df-rex 3070 . . 3 (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹𝐵) ∧ 𝜑))
2 fvelimab 6979 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
32anbi1d 631 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑥 ∈ (𝐹𝐵) ∧ 𝜑) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑)))
43exbidv 1921 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥(𝑥 ∈ (𝐹𝐵) ∧ 𝜑) ↔ ∃𝑥(∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑)))
51, 4bitrid 283 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑥(∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑)))
6 rexcom4 3287 . . 3 (∃𝑦𝐵𝑥((𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑥𝑦𝐵 ((𝐹𝑦) = 𝑥𝜑))
7 eqcom 2743 . . . . . . 7 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
87anbi1i 624 . . . . . 6 (((𝐹𝑦) = 𝑥𝜑) ↔ (𝑥 = (𝐹𝑦) ∧ 𝜑))
98exbii 1848 . . . . 5 (∃𝑥((𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑥(𝑥 = (𝐹𝑦) ∧ 𝜑))
10 fvex 6917 . . . . . 6 (𝐹𝑦) ∈ V
11 imaeqsexvOLD.1 . . . . . 6 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
1210, 11ceqsexv 3531 . . . . 5 (∃𝑥(𝑥 = (𝐹𝑦) ∧ 𝜑) ↔ 𝜓)
139, 12bitri 275 . . . 4 (∃𝑥((𝐹𝑦) = 𝑥𝜑) ↔ 𝜓)
1413rexbii 3093 . . 3 (∃𝑦𝐵𝑥((𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑦𝐵 𝜓)
15 r19.41v 3188 . . . 4 (∃𝑦𝐵 ((𝐹𝑦) = 𝑥𝜑) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑))
1615exbii 1848 . . 3 (∃𝑥𝑦𝐵 ((𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑥(∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑))
176, 14, 163bitr3ri 302 . 2 (∃𝑥(∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑦𝐵 𝜓)
185, 17bitrdi 287 1 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  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:  imaeqsalvOLD  7382
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