Theorem rexeqf

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003) (Revised by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 9-Mar-2025)

	Ref	Expression
Hypotheses	raleqf.1	⊢ Ⅎ 𝑥 𝐴
	raleqf.2	⊢ Ⅎ 𝑥 𝐵
Assertion	rexeqf	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		raleqf.1	⊢ Ⅎ 𝑥 𝐴
2		raleqf.2	⊢ Ⅎ 𝑥 𝐵
3	1 2	raleqf	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝜑 ) )
4		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜑 )
5		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐵 𝜑 )
6	3 4 5	3bitr3g	⊢ ( 𝐴 = 𝐵 → ( ¬ ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃ 𝑥 ∈ 𝐵 𝜑 ) )
7	6	con4bid	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) )