Theorem rexeqbidvv 3308

Description: Version of rexeqbidv 3306 with additional disjoint variable conditions, not requiring ax-8 2116 nor df-clel 2812. (Contributed by Wolf Lammen, 25-Sep-2024.)

Hypotheses

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

Assertion

Ref	Expression
rexeqbidvv	⊢ (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 rexeqbidvv

Step	Hyp	Ref	Expression
1		raleqbidvv.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbidvv.2	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	notbid 321	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
4	1, 3	raleqbidvv 3307	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑥 ∈ 𝐵 ¬ 𝜒))
5		ralnex 3150	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
6		ralnex 3150	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜒 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜒)
7	4, 5, 6	3bitr3g 316	. 2 ⊢ (𝜑 → (¬ ∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐵 𝜒))
8	7	con4bid 320	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1542  ∀wral 3054  ∃wrex 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-cleq 2731  df-ral 3059  df-rex 3060

This theorem is referenced by:  rexeqbi1dv  3310