Theorem 2alimi 1845

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 1844	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1844	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-gen 1828  ax-4 1842

This theorem is used by:  alcomimw  2076  2mo  2679  2eu6  2687  euind  3690  reuind  3719  sbnfc2  4407  opelopabt  5521  ssrel  5774  ssrelrel  5787  fundif  6592  opabbrex  7476  fnoprabg  7546  tz7.48lem  8437  ssrelf  32997  bj-3exbi  37266  mpobi123f  38852  mptbi12f  38856  ismrc  43473  refimssco  44374  19.33-2  45133  pm11.63  45146  pm11.71  45148  axc5c4c711to11  45156  ichal  48256