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Metamath Proof Explorer

Theorem rexeqif

Description: Equality inference for restricted existential quantifier. (Contributed by Glauco Siliprandi, 15-Feb-2025)

Ref Expression
Hypotheses rexeqif.1 ⊢ Ⅎ 𝑥 𝐴
rexeqif.2 ⊢ Ⅎ 𝑥 𝐵
rexeqif.3 ⊢ 𝐴 = 𝐵
Assertion rexeqif ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 )

Proof

Step Hyp Ref Expression
1 rexeqif.1 ⊢ Ⅎ 𝑥 𝐴
2 rexeqif.2 ⊢ Ⅎ 𝑥 𝐵
3 rexeqif.3 ⊢ 𝐴 = 𝐵
4 1 2 rexeqf ⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) )
5 3 4 ax-mp ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 )