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  2643  2eu6  2652  euind  3683  reuind  3712  sbnfc2  4389  opelopabt  5472  ssrel  5723  ssrelrel  5736  fundif  6530  opabbrex  7399  fnoprabg  7469  tz7.48lem  8360  ssrelf  32596  bj-3exbi  36656  mpobi123f  38208  mptbi12f  38212  ismrc  42740  refimssco  43646  19.33-2  44421  pm11.63  44434  pm11.71  44436  axc5c4c711to11  44444  ichal  47503