Theorem 2alimi 1907

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 1906	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1906	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1650

This theorem was proved from axioms:  ax-mp 5  ax-gen 1890  ax-4 1904

This theorem is referenced by:  alcomiw  2138  2mo  2673  2eu6  2680  euind  3554  reuind  3574  sbnfc2  4171  opelopabt  5150  ssrel  5379  ssrelrel  5391  fundif  6118  opabbrex  6897  fnoprabg  6963  tz7.48lem  7744  ssrelf  29896  bj-3exbi  33057  bj-mo3OLD  33282  mpt2bi123f  34413  mptbi12f  34417  ismrc  37966  refimssco  38612  19.33-2  39279  pm11.63  39293  pm11.71  39295  axc5c4c711to11  39303