Theorem 2alimi 1813

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 1812	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1812	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-gen 1796  ax-4 1810

This theorem is referenced by:  alcomimw  2044  2mo  2648  2eu6  2657  euind  3682  reuind  3711  sbnfc2  4391  opelopabt  5480  ssrel  5732  ssrelrel  5745  fundif  6541  opabbrex  7411  fnoprabg  7481  tz7.48lem  8372  ssrelf  32693  bj-3exbi  36816  mpobi123f  38363  mptbi12f  38367  ismrc  42943  refimssco  43848  19.33-2  44623  pm11.63  44636  pm11.71  44638  axc5c4c711to11  44646  ichal  47712