Theorem bj-biexal3

Description: When ph is substituted for ps , both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019)

	Ref	Expression
Assertion	bj-biexal3	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-biexal1	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
2		bj-biexal2	⊢ ( ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
3	1 2	bitr4i	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜓 ) )