Theorem 2alimi 1835

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 1834	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1834	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1561

This theorem was proved from axioms:  ax-mp 5  ax-gen 1818  ax-4 1832

This theorem is referenced by:  alcomimw  2066  2mo  2678  2eu6  2686  euind  3690  reuind  3719  sbnfc2  4396  opelopabt  5506  ssrel  5759  ssrelrel  5772  fundif  6574  opabbrex  7453  fnoprabg  7523  tz7.48lem  8416  ssrelf  32868  bj-3exbi  37082  mpobi123f  38668  mptbi12f  38672  ismrc  43289  refimssco  44190  19.33-2  44951  pm11.63  44964  pm11.71  44966  axc5c4c711to11  44974  ichal  48071