Theorem bj-nnflemee 34113

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.)

Assertion

Ref	Expression
bj-nnflemee	⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑))

Proof of Theorem bj-nnflemee

Step	Hyp	Ref	Expression
1		excom 2168	. 2 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
2		exim 1833	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑥∃𝑦𝜑 → ∃𝑥𝜑))
3	1, 2	syl5bi 244	1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-11 2160

This theorem depends on definitions:  df-bi 209  df-ex 1780

This theorem is referenced by:  bj-nnfext  34117