Theorem bj-cbv3ta

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

	Ref	Expression
Assertion	bj-cbv3ta	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-spimt2	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )
2	1	imp	⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ∧ ( ∃ 𝑥 𝜓 → 𝜓 ) ) → ( ∀ 𝑥 𝜑 → 𝜓 ) )
3	2	alanimi	⊢ ( ( ∀ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ∧ ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) ) → ∀ 𝑦 ( ∀ 𝑥 𝜑 → 𝜓 ) )
4		bj-hbalt	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
5		sylgt	⊢ ( ∀ 𝑦 ( ∀ 𝑥 𝜑 → 𝜓 ) → ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
6	3 4 5	syl2im	⊢ ( ( ∀ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) ∧ ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) ) → ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
7	6	expimpd	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )
8	7	alcoms	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) → ( ( ∀ 𝑦 ( ∃ 𝑥 𝜓 → 𝜓 ) ∧ ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ) )