Theorem 2alimi 1831

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 1830	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1830	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-gen 1814  ax-4 1828

This theorem is referenced by:  alcomimw  2062  2mo  2674  2eu6  2682  euind  3685  reuind  3714  sbnfc2  4390  opelopabt  5499  ssrel  5751  ssrelrel  5764  fundif  6564  opabbrex  7443  fnoprabg  7513  tz7.48lem  8405  ssrelf  32777  bj-3exbi  37035  mpobi123f  38621  mptbi12f  38625  ismrc  43242  refimssco  44143  19.33-2  44918  pm11.63  44931  pm11.71  44933  axc5c4c711to11  44941  ichal  48032