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Theorem ralimi2 2492
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
Hypothesis
Ref Expression
ralimi2.1 ((𝑥𝐴𝜑) → (𝑥𝐵𝜓))
Assertion
Ref Expression
ralimi2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)

Proof of Theorem ralimi2
StepHypRef Expression
1 ralimi2.1 . . 3 ((𝑥𝐴𝜑) → (𝑥𝐵𝜓))
21alimi 1431 . 2 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐵𝜓))
3 df-ral 2421 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 2421 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
52, 3, 43imtr4i 200 1 (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  wcel 1480  wral 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425
This theorem depends on definitions:  df-bi 116  df-ral 2421
This theorem is referenced by:  ralimia  2493  ralcom3  2598  bj-nntrans  13179  bj-findis  13207
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