Theorem raleq

Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025)

	Ref	Expression
Assertion	raleq	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		rexeq	⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 ¬ 𝜑 ) )
2		rexnal	⊢ ( ∃ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 𝜑 )
3		rexnal	⊢ ( ∃ 𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐵 𝜑 )
4	1 2 3	3bitr3g	⊢ ( 𝐴 = 𝐵 → ( ¬ ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐵 𝜑 ) )
5	4	con4bid	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 𝜑 ) )