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Theorem ral2imi 3085
Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3086. (Revised by Wolf Lammen, 1-Dec-2019.)
Hypothesis
Ref Expression
ral2imi.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ral2imi (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ral2imi
StepHypRef Expression
1 df-ral 3062 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 ral2imi.1 . . . . 5 (𝜑 → (𝜓𝜒))
32imim3i 64 . . . 4 ((𝑥𝐴𝜑) → ((𝑥𝐴𝜓) → (𝑥𝐴𝜒)))
43al2imi 1817 . . 3 (∀𝑥(𝑥𝐴𝜑) → (∀𝑥(𝑥𝐴𝜓) → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 3062 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
6 df-ral 3062 . . 3 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
74, 5, 63imtr4g 295 . 2 (∀𝑥(𝑥𝐴𝜑) → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
81, 7sylbi 216 1 (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539  wcel 2106  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 206  df-ral 3062
This theorem is referenced by:  ralim  3086  rexim  3087  ralbi  3103  r19.26  3111  iiner  8785  ss2ixp  8906  undifixp  8930  boxriin  8936  acni2  10043  axcc4  10436  intgru  10811  ingru  10812  prdsdsval3  17435  mndind  18745  hauscmplem  23130  uspgr2wlkeq  29158  wlkp1lem8  29192  prdstotbnd  36965  mnuunid  43338
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