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Metamath Proof Explorer

Theorem 19.23vv

Description: Theorem 19.23v extended to two variables. (Contributed by NM, 10-Aug-2004)

Ref Expression
Assertion 19.23vv ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 19.23v ⊢ ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑦 𝜑 → 𝜓 ) )
2 1 albii ⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜓 ) )
3 19.23v ⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 → 𝜓 ) )
4 2 3 bitri ⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 → 𝜓 ) )