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  2648  2eu6  2657  euind  3670  reuind  3699  sbnfc2  4379  opelopabt  5487  ssrel  5739  ssrelrel  5752  fundif  6547  opabbrex  7420  fnoprabg  7490  tz7.48lem  8380  ssrelf  32688  bj-3exbi  36897  mpobi123f  38483  mptbi12f  38487  ismrc  43133  refimssco  44034  19.33-2  44809  pm11.63  44822  pm11.71  44824  axc5c4c711to11  44832  ichal  47926