Theorem alcoms 2157

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 2156	. 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 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-11 2156

This theorem is referenced by:  cbv2h  2406  mo3  2564  bj-nfalt  34820  bj-cbv3ta  34895  bj-cbv2hv  34906  wl-equsal1i  35629  wl-mo3t  35658  axc11n-16  36879  axc11next  41913