Theorem bj-cbv3tb

Description: Closed form of cbv3 . (Contributed by BJ, 2-May-2019)

	Ref	Expression
Assertion	bj-cbv3tb	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑦 Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 Ⅎ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.9t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∃ 𝑥 𝜓 ↔ 𝜓 ) )
2	1	biimpd	⊢ ( Ⅎ 𝑥 𝜓 → ( ∃ 𝑥 𝜓 → 𝜓 ) )
3	2	alimi	⊢ ( ∀ 𝑦 Ⅎ 𝑥 𝜓 → ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) )
4		nf5r	⊢ ( Ⅎ 𝑦 𝜑 → ( 𝜑 → ∀ 𝑦 𝜑 ) )
5	4	alimi	⊢ ( ∀ 𝑥 Ⅎ 𝑦 𝜑 → ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) )
6		bj-cbv3ta	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
7	3 5 6	syl2ani	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑦 Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 Ⅎ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )