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Theorem ralimi2 3098
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 1887 . 2 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐵𝜓))
3 df-ral 3066 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 3066 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
52, 3, 43imtr4i 281 1 (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wcel 2145  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885
This theorem depends on definitions:  df-bi 197  df-ral 3066
This theorem is referenced by:  ralimia  3099  ralcom3  3253  tfi  7204  resixpfo  8104  omex  8708  kmlem1  9178  brdom5  9557  brdom4  9558  xrub  12347  pcmptcl  15802  itgeq2  23764  iblcnlem  23775  pntrsumbnd  25476  nmounbseqi  27972  nmounbseqiALT  27973  sumdmdi  29619  dmdbr4ati  29620  dmdbr6ati  29622  bnj110  31266  fiinfi  38402
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