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Metamath Proof Explorer

Theorem 19.26-2

Description: Theorem 19.26 with two quantifiers. (Contributed by NM, 3-Feb-2005)

Ref Expression
Assertion 19.26-2 ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) )

Proof

Step Hyp Ref Expression
1 19.26 ⊢ ( ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) )
2 1 albii ⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ∀ 𝑥 ( ∀ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) )
3 19.26 ⊢ ( ∀ 𝑥 ( ∀ 𝑦 𝜑 ∧ ∀ 𝑦 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) )
4 2 3 bitri ⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∧ ∀ 𝑥 ∀ 𝑦 𝜓 ) )