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Metamath Proof Explorer

Theorem r19.21t

Description: Restricted quantifier version of 19.21t ; closed form of r19.21 . (Contributed by NM, 1-Mar-2008) (Proof shortened by Wolf Lammen, 2-Jan-2020)

		Ref	Expression
	Assertion	r19.21t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.21t	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
3		bi2.04	⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) ↔ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
4	3	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) ↔ ∀ 𝑥 ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
5	2 4	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
6		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
7	6	imbi2i	⊢ ( ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
8	1 5 7	3bitr4g	⊢ ( Ⅎ 𝑥 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 ) ) )