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Metamath Proof Explorer

Theorem 2exbii

Description: Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995)

Ref Expression
Hypothesis 2exbii.1 ⊢ ( 𝜑 ↔ 𝜓 )
Assertion 2exbii ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 𝜓 )

Proof

Step Hyp Ref Expression
1 2exbii.1 ⊢ ( 𝜑 ↔ 𝜓 )
2 1 exbii ⊢ ( ∃ 𝑦 𝜑 ↔ ∃ 𝑦 𝜓 )
3 2 exbii ⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 𝜓 )