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Metamath Proof Explorer

Theorem bj-biexal1

Description: A general FOL biconditional that generalizes 19.9ht among others. For this and the following theorems, see also 19.35 , 19.21 , 19.23 . When ph is substituted for ps , both sides express a form of nonfreeness. (Contributed by BJ, 20-Oct-2019)

		Ref	Expression
	Assertion	bj-biexal1	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 𝜓
2	1	19.23	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )