Theorem 2alimi 1509

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 1508	. 2 ⊢ (∀𝑦𝜑 → ∀𝑦𝜓)
3	2	alimi 1508	1 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1400

This proof depends on axioms:  ax-mp 5  ax-5 1500  ax-gen 1502

This theorem is used by:  mo23  2128  mo3h  2140  spc2gv  2916  spc3gv  2918  euind  3013  reuind  3031  sbnfc2  3208  opelopabt  4404  ssrel  4863  ssrelrel  4875  fundif  5425  fnoprabg  6189  wexmiddifxy  17058