Theorem bj-nnflemaa 37470

Description: One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 37364. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-nnflemaa

Step	Hyp	Ref	Expression
1		alim 1843	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑥∀𝑦𝜓))
2		ax-11 2195	. 2 ⊢ (∀𝑥∀𝑦𝜓 → ∀𝑦∀𝑥𝜓)
3	1, 2	syl6 36	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))

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

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-4 1842  ax-11 2195

This theorem is used by:  bj-nnfalt  37474