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Metamath Proof Explorer

Theorem exbii

Description: Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994)

Ref Expression
Hypothesis exbii.1 ⊢ ( 𝜑 ↔ 𝜓 )
Assertion exbii ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 )

Proof

Step Hyp Ref Expression
1 exbii.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 exbi ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 ) )
3 2 1 mpg ⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 𝜓 )