Theorem bj-19.21t0

Description: Proof of 19.21t from stdpc5t . (Contributed by BJ, 15-Sep-2018) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-19.21t0	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		stdpc5t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜓 ) ) )
2		19.9t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 ↔ 𝜑 ) )
3	2	imbi1d	⊢ ( Ⅎ 𝑥 𝜑 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )
4		19.38	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) )
5	3 4	syl6bir	⊢ ( Ⅎ 𝑥 𝜑 → ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
6	1 5	impbid	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 𝜓 ) ) )