Theorem alcoms 2158

Description: Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)

Hypothesis

Ref	Expression
alcoms.1	⊢ (∀𝑥∀𝑦𝜑 → 𝜓)

Assertion

Ref	Expression
alcoms	⊢ (∀𝑦∀𝑥𝜑 → 𝜓)

Proof of Theorem alcoms

Step	Hyp	Ref	Expression
1		ax-11 2157	. 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
2		alcoms.1	. 2 ⊢ (∀𝑥∀𝑦𝜑 → 𝜓)
3	1, 2	syl 17	1 ⊢ (∀𝑦∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-11 2157

This theorem is referenced by:  cbv2h  2411  mo3  2564  bj-nfalt  36712  bj-cbv3ta  36787  bj-cbv2hv  36798  wl-equsal1i  37545  wl-mo3t  37577  axc11n-16  38939  axc11next  44425