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Theorem imaeqsexvOLD 7382
Description: Obsolete version of rexima 7257 as of 14-Aug-2025. Duplicate version of rexima 7257. (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 3068 . . 3 (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐹𝐵) ∧ 𝜑))
2 fvelimab 6980 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑥 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑥))
32anbi1d 631 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑥 ∈ (𝐹𝐵) ∧ 𝜑) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑)))
43exbidv 1918 . . 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 3285 . . 3 (∃𝑦𝐵𝑥((𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑥𝑦𝐵 ((𝐹𝑦) = 𝑥𝜑))
7 eqcom 2741 . . . . . . 7 ((𝐹𝑦) = 𝑥𝑥 = (𝐹𝑦))
87anbi1i 624 . . . . . 6 (((𝐹𝑦) = 𝑥𝜑) ↔ (𝑥 = (𝐹𝑦) ∧ 𝜑))
98exbii 1844 . . . . 5 (∃𝑥((𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑥(𝑥 = (𝐹𝑦) ∧ 𝜑))
10 fvex 6919 . . . . . 6 (𝐹𝑦) ∈ V
11 imaeqsexvOLD.1 . . . . . 6 (𝑥 = (𝐹𝑦) → (𝜑𝜓))
1210, 11ceqsexv 3529 . . . . 5 (∃𝑥(𝑥 = (𝐹𝑦) ∧ 𝜑) ↔ 𝜓)
139, 12bitri 275 . . . 4 (∃𝑥((𝐹𝑦) = 𝑥𝜑) ↔ 𝜓)
1413rexbii 3091 . . 3 (∃𝑦𝐵𝑥((𝐹𝑦) = 𝑥𝜑) ↔ ∃𝑦𝐵 𝜓)
15 r19.41v 3186 . . . 4 (∃𝑦𝐵 ((𝐹𝑦) = 𝑥𝜑) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑥𝜑))
1615exbii 1844 . . 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 1536  wex 1775  wcel 2105  wrex 3067  wss 3962  cima 5691   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  imaeqsalvOLD  7383
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