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Theorem ral2imi 3084
Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 3085. (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 1816 . . 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 1538  wcel 2105  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810
This theorem depends on definitions:  df-bi 206  df-ral 3062
This theorem is referenced by:  ralim  3085  rexim  3086  ralbi  3102  r19.26  3110  iiner  8649  ss2ixp  8769  undifixp  8793  boxriin  8799  acni2  9903  axcc4  10296  intgru  10671  ingru  10672  prdsdsval3  17293  mndind  18563  hauscmplem  22663  uspgr2wlkeq  28302  wlkp1lem8  28336  prdstotbnd  36057  mnuunid  42216
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