Theorem bj-nnflemae 37211

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

Assertion

Ref	Expression
bj-nnflemae	⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜓))

Proof of Theorem bj-nnflemae

Step	Hyp	Ref	Expression
1		exim 1848	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∃𝑥∀𝑦𝜓))
2		bj-19.12 37146	. 2 ⊢ (∃𝑥∀𝑦𝜓 → ∀𝑦∃𝑥𝜓)
3	1, 2	syl6 35	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1552  ∃wex 1793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-10 2169  ax-11 2185  ax-12 2206

This theorem depends on definitions:  df-bi 209  df-ex 1794

This theorem is referenced by:  bj-nnfext  37214