Theorem 2alimi 1812

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 1811	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1811	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-gen 1795  ax-4 1809

This theorem is referenced by:  alcomimw  2043  2mo  2641  2eu6  2650  euind  3686  reuind  3715  sbnfc2  4392  opelopabt  5479  ssrel  5730  ssrelrel  5743  fundif  6535  opabbrex  7406  fnoprabg  7476  tz7.48lem  8370  ssrelf  32576  bj-3exbi  36592  mpobi123f  38144  mptbi12f  38148  ismrc  42677  refimssco  43583  19.33-2  44358  pm11.63  44371  pm11.71  44373  axc5c4c711to11  44381  ichal  47454