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Metamath Proof Explorer

Theorem rexeqi

Description: Equality inference for restricted existential quantifier. (Contributed by Mario Carneiro, 23-Apr-2015)

Ref Expression
Hypothesis raleq1i.1 ⊢ 𝐴 = 𝐵
Assertion rexeqi ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 )

Proof

Step Hyp Ref Expression
1 raleq1i.1 ⊢ 𝐴 = 𝐵
2 rexeq ⊢ ( 𝐴 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 ) )
3 1 2 ax-mp ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ∈ 𝐵 𝜑 )