Theorem bj-nnflemee 37659

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 → 𝜓) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem bj-nnflemee

Step	Hyp	Ref	Expression
1		excom 2199	. 2 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
2		exim 1867	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑥∃𝑦𝜑 → ∃𝑥𝜓))
3	1, 2	biimtrid 245	1 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-11 2194

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bj-nnfext  37663