Theorem bj-nnflemee

Description: One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-nnflemee	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜓 ) → ( ∃ 𝑦 ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		excom	⊢ ( ∃ 𝑦 ∃ 𝑥 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 𝜑 )
2		exim	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑥 𝜓 ) )
3	1 2	biimtrid	⊢ ( ∀ 𝑥 ( ∃ 𝑦 𝜑 → 𝜓 ) → ( ∃ 𝑦 ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )