Theorem 2alimi 1842

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 1841	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1841	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-gen 1825  ax-4 1839

This theorem is referenced by:  alcomimw  2073  2mo  2676  2eu6  2684  euind  3688  reuind  3717  sbnfc2  4405  opelopabt  5518  ssrel  5771  ssrelrel  5784  fundif  6587  opabbrex  7465  fnoprabg  7535  tz7.48lem  8429  ssrelf  32941  bj-3exbi  37206  mpobi123f  38792  mptbi12f  38796  ismrc  43415  refimssco  44316  19.33-2  45075  pm11.63  45088  pm11.71  45090  axc5c4c711to11  45098  ichal  48198