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Theorem ralimi2 3076
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 1808 . 2 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐵𝜓))
3 df-ral 3060 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
4 df-ral 3060 . 2 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
52, 3, 43imtr4i 292 1 (∀𝑥𝐴 𝜑 → ∀𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-ral 3060
This theorem is referenced by:  ralimia  3078  ralcom3OLD  3096  tfi  7874  resixpfo  8975  omex  9681  kmlem1  10189  brdom5  10567  brdom4  10568  xrub  13351  pcmptcl  16925  itgeq2  25828  iblcnlem  25839  pntrsumbnd  27625  nmounbseqi  30806  nmounbseqiALT  30807  sumdmdi  32449  dmdbr4ati  32450  dmdbr6ati  32452  bnj110  34851  fiinfi  43563
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