Theorem bj-nnflemaa 34112

Description: One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 34036. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-nnflemaa

Step	Hyp	Ref	Expression
1		alim 1810	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑥∀𝑦𝜑))
2		ax-11 2160	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
3	1, 2	syl6 35	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1534

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

This theorem is referenced by:  bj-nnfalt  34116