Theorem imaeqsalvOLD 7312

Description: Obsolete version of ralima 7185 as of 14-Aug-2025. Duplicate version of ralima 7185. (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 7311	. . 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 3088	. 2 ⊢ (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ¬ ∃𝑥 ∈ (𝐹 “ 𝐵) ¬ 𝜑)
6		dfral2 3088	. 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 1542  ∀wral 3052  ∃wrex 3061   ⊆ wss 3902   “ cima 5628   Fn wfn 6488  ‘cfv 6493

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501

This theorem is referenced by: (None)