Theorem imaeqsalvOLD 7383

Description: Obsolete version of ralima 7256 as of 14-Aug-2025. Duplicate version of ralima 7256. (Contributed by Scott Fenton, 27-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
imaeqsexvOLD.1	⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
imaeqsalvOLD	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 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

Step	Hyp	Ref	Expression
1		imaeqsexvOLD.1	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
2	1	notbid 318	. . . 4 ⊢ (𝑥 = (𝐹‘𝑦) → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	imaeqsexvOLD 7382	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
4	3	notbid 318	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (¬ ∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑 ↔ ¬ ∃𝑦 ∈ 𝐵 ¬ 𝜓))
5		dfral2 3096	. 2 ⊢ (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ¬ ∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑)
6		dfral2 3096	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∃𝑦 ∈ 𝐵 ¬ 𝜓)
7	4, 5, 6	3bitr4g 314	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ∀wral 3058  ∃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: (None)