Theorem bj-nnflemee 34872

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 2164	. 2 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
2		exim 1837	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑥∃𝑦𝜑 → ∃𝑥𝜑))
3	1, 2	syl5bi 241	1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-11 2156

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  bj-nnfext  34876