Theorem rexeqbidvv 3306

Description: Version of rexeqbidv 3314 with additional disjoint variable conditions, not requiring ax-8 2121 nor df-clel 2814. (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	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbidvv.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rexeqbidva 3304	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  ∃wrex 3063

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 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-rex 3064

This theorem is referenced by:  rexeqbi1dv  3308  constrsuc  33922