Theorem 2alimi 1819

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 1818	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1818	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-gen 1802  ax-4 1816

This theorem is referenced by:  alcomimw  2050  2mo  2652  2eu6  2661  euind  3672  reuind  3701  sbnfc2  4374  opelopabt  5481  ssrel  5733  ssrelrel  5746  fundif  6541  opabbrex  7416  fnoprabg  7486  tz7.48lem  8377  ssrelf  32714  bj-3exbi  36950  mpobi123f  38536  mptbi12f  38540  ismrc  43157  refimssco  44058  19.33-2  44833  pm11.63  44846  pm11.71  44848  axc5c4c711to11  44856  ichal  47948