Theorem 2alimi 1731

Description: Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)

Hypothesis

Ref	Expression
alimi.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
2alimi	⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Proof of Theorem 2alimi

Step	Hyp	Ref	Expression
1		alimi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	alimi 1730	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1730	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-gen 1713  ax-4 1728

This theorem is referenced by:  2mo  2539  2eu6  2546  euind  3360  reuind  3378  sbnfc2  3959  opelopabt  4912  ssrel  5130  ssrelOLD  5131  ssrelrel  5143  fundif  5849  opabbrex  6593  fnoprabg  6659  tz7.48lem  7423  ssrelf  28805  bj-3exbi  31785  bj-mo3OLD  32022  mpt2bi123f  33141  mptbi12f  33145  ismrc  36282  refimssco  36932  19.33-2  37603  pm11.63  37617  pm11.71  37619  axc5c4c711to11  37628