Theorem 19.12vvv

Description: Version of 19.12vv with a disjoint variable condition, requiring fewer axioms. See also 19.12 . (Contributed by BJ, 18-Mar-2020)

	Ref	Expression
Assertion	19.12vvv	⊢ ( ∃ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.21v	⊢ ( ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → ∀ 𝑦 𝜓 ) )
2	1	exbii	⊢ ( ∃ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) )
3		19.36v	⊢ ( ∃ 𝑥 ( 𝜑 → ∀ 𝑦 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) )
4		19.36v	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → 𝜓 ) )
5	4	albii	⊢ ( ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑦 ( ∀ 𝑥 𝜑 → 𝜓 ) )
6		19.21v	⊢ ( ∀ 𝑦 ( ∀ 𝑥 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) )
7	5 6	bitr2i	⊢ ( ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) ↔ ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) )
8	2 3 7	3bitri	⊢ ( ∃ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑦 ∃ 𝑥 ( 𝜑 → 𝜓 ) )