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  2078  2mo  2676  2eu6  2684  euind  3685  reuind  3714  sbnfc2  4400  opelopabt  5511  ssrel  5765  ssrelrel  5778  fundifOLD  6614  opabbrex  7499  fnoprabg  7569  tz7.48lemOLD  8482  ssrelf  33284  bj-3exbi  37693  mpobi123f  39297  mptbi12f  39301  ismrc  43916  refimssco  44810  19.33-2  45569  pm11.63  45582  pm11.71  45584  axc5c4c711to11  45592  ichal  48763