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  2674  2eu6  2682  euind  3682  reuind  3711  sbnfc2  4397  opelopabt  5506  ssrel  5759  ssrelrel  5772  fundif  6581  opabbrex  7465  fnoprabg  7535  tz7.48lemOLD  8435  ssrelf  33191  bj-3exbi  37472  mpobi123f  39062  mptbi12f  39066  ismrc  43665  refimssco  44566  19.33-2  45325  pm11.63  45338  pm11.71  45340  axc5c4c711to11  45348  ichal  48492