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  3684  reuind  3713  sbnfc2  4393  opelopabt  5488  ssrel  5740  ssrelrel  5753  fundif  6549  opabbrex  7421  fnoprabg  7491  tz7.48lem  8382  ssrelf  32704  bj-3exbi  36852  mpobi123f  38410  mptbi12f  38414  ismrc  43055  refimssco  43960  19.33-2  44735  pm11.63  44748  pm11.71  44750  axc5c4c711to11  44758  ichal  47823