This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem bj-pm11.53v

Description: Version of pm11.53v with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024)

Ref Expression
Assertion bj-pm11.53v ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ ∀ 𝑦 Ⅎ' 𝑥 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 bj-nnfalt ⊢ ( ∀ 𝑦 Ⅎ' 𝑥 𝜓 → Ⅎ' 𝑥 ∀ 𝑦 𝜓 )
2 bj-pm11.53vw ⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ Ⅎ' 𝑥 ∀ 𝑦 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
3 1 2 sylan2 ⊢ ( ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ∧ ∀ 𝑦 Ⅎ' 𝑥 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )