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  2645  2eu6  2654  euind  3679  reuind  3708  sbnfc2  4388  opelopabt  5477  ssrel  5729  ssrelrel  5742  fundif  6537  opabbrex  7407  fnoprabg  7477  tz7.48lem  8368  ssrelf  32602  bj-3exbi  36683  mpobi123f  38225  mptbi12f  38229  ismrc  42821  refimssco  43727  19.33-2  44502  pm11.63  44515  pm11.71  44517  axc5c4c711to11  44525  ichal  47593