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

Metamath Proof Explorer

Theorem bj-nnfalt

Description: See nfal and bj-nfalt . (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

Ref Expression
Assertion bj-nnfalt ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 → Ⅎ' 𝑦 ∀ 𝑥 𝜑 )

Proof

Step Hyp Ref Expression
1 df-bj-nnf ⊢ ( Ⅎ' 𝑦 𝜑 ↔ ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) )
2 1 albii ⊢ ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 ↔ ∀ 𝑥 ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) )
3 simpl ⊢ ( ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( ∃ 𝑦 𝜑 → 𝜑 ) )
4 3 alimi ⊢ ( ∀ 𝑥 ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜑 ) )
5 bj-nnflemea ⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜑 ) → ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
6 4 5 syl ⊢ ( ∀ 𝑥 ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
7 2 6 sylbi ⊢ ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 → ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
8 simpr ⊢ ( ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( 𝜑 → ∀ 𝑦 𝜑 ) )
9 8 alimi ⊢ ( ∀ 𝑥 ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) )
10 bj-nnflemaa ⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
11 9 10 syl ⊢ ( ∀ 𝑥 ( ( ∃ 𝑦 𝜑 → 𝜑 ) ∧ ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
12 2 11 sylbi ⊢ ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
13 df-bj-nnf ⊢ ( Ⅎ' 𝑦 ∀ 𝑥 𝜑 ↔ ( ( ∃ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ∧ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) ) )
14 7 12 13 sylanbrc ⊢ ( ∀ 𝑥 Ⅎ' 𝑦 𝜑 → Ⅎ' 𝑦 ∀ 𝑥 𝜑 )