Theorem rexeqbidvvOLD 3327

Description: Obsolete version of rexeqbidvv 3326 as of 9-Mar-2025. (Contributed by Wolf Lammen, 25-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
raleqbidvv.1	⊢ (𝜑 → 𝐴 = 𝐵)
raleqbidvv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexeqbidvvOLD	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rexeqbidvvOLD

Step	Hyp	Ref	Expression
1		raleqbidvv.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbidvv.2	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	notbid 318	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
4	1, 3	raleqbidvv 3324	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝜒))
5		ralnex 3067	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
6		ralnex 3067	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜒 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜒)
7	4, 5, 6	3bitr3g 313	. 2 ⊢ (𝜑 → (¬ ∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜒))
8	7	con4bid 317	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1534  ∀wral 3056  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2719  df-ral 3057  df-rex 3066

This theorem is referenced by: (None)