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Theorem ral2imi 2946
Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 2947. (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 2916 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 ral2imi.1 . . . . 5 (𝜑 → (𝜓𝜒))
32imim3i 64 . . . 4 ((𝑥𝐴𝜑) → ((𝑥𝐴𝜓) → (𝑥𝐴𝜒)))
43al2imi 1742 . . 3 (∀𝑥(𝑥𝐴𝜑) → (∀𝑥(𝑥𝐴𝜓) → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2916 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
6 df-ral 2916 . . 3 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
74, 5, 63imtr4g 285 . 2 (∀𝑥(𝑥𝐴𝜑) → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
81, 7sylbi 207 1 (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1480  wcel 1989  wral 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736
This theorem depends on definitions:  df-bi 197  df-ral 2916
This theorem is referenced by:  ralim  2947  rexim  3007  r19.26  3062  iiner  7816  ss2ixp  7918  undifixp  7941  boxriin  7947  acni2  8866  axcc4  9258  intgru  9633  ingru  9634  prdsdsval3  16139  mrcmndind  17360  hauscmplem  21203  uspgr2wlkeq  26536  wlkp1lem8  26571  prdstotbnd  33573
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