Theorem 2alimi 1845

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 1844	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1844	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-gen 1828  ax-4 1842

This theorem is used by:  alcomimw  2076  2mo  2675  2eu6  2683  euind  3685  reuind  3714  sbnfc2  4400  opelopabt  5514  ssrel  5767  ssrelrel  5780  fundif  6586  opabbrex  7470  fnoprabg  7540  tz7.48lem  8434  ssrelf  33096  bj-3exbi  37341  mpobi123f  38918  mptbi12f  38922  ismrc  43554  refimssco  44455  19.33-2  45214  pm11.63  45227  pm11.71  45229  axc5c4c711to11  45237  ichal  48374