Theorem 2alimi 1814

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 1813	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1813	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-gen 1797  ax-4 1811

This theorem is referenced by:  alcomimw  2045  2mo  2649  2eu6  2658  euind  3671  reuind  3700  sbnfc2  4380  opelopabt  5480  ssrel  5732  ssrelrel  5745  fundif  6541  opabbrex  7413  fnoprabg  7483  tz7.48lem  8373  ssrelf  32703  bj-3exbi  36913  mpobi123f  38497  mptbi12f  38501  ismrc  43147  refimssco  44052  19.33-2  44827  pm11.63  44840  pm11.71  44842  axc5c4c711to11  44850  ichal  47938