Theorem rexeqbidv

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007) Remove usage of ax-10 , ax-11 , and ax-12 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023)

	Ref	Expression
Hypotheses	raleqbidv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	raleqbidv.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	rexeqbidv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		raleqbidv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		raleqbidv.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	3 2	anbi12d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
5	4	rexbidv2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 ) )