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Metamath Proof Explorer

Theorem bj-19.21t

Description: Statement 19.21t proved from modalK (obsoleting 19.21v ). (Contributed by BJ, 2-Dec-2023)

Ref Expression
Assertion bj-19.21t ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 bj-nnf-alrim ⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) ) )
2 bj-nnfe ⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜑 ) )
3 2 imim1d ⊢ ( Ⅎ' 𝑥 𝜑 → ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ) )
4 19.38 ⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
5 3 4 syl6 ⊢ ( Ⅎ' 𝑥 𝜑 → ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
6 1 5 impbid ⊢ ( Ⅎ' 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )