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Theorem ral2imi 2976
Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) Allow shortening of ralim 2977. (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 2946 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
2 ral2imi.1 . . . . 5 (𝜑 → (𝜓𝜒))
32imim3i 64 . . . 4 ((𝑥𝐴𝜑) → ((𝑥𝐴𝜓) → (𝑥𝐴𝜒)))
43al2imi 1783 . . 3 (∀𝑥(𝑥𝐴𝜑) → (∀𝑥(𝑥𝐴𝜓) → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2946 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
6 df-ral 2946 . . 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 1521  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777
This theorem depends on definitions:  df-bi 197  df-ral 2946
This theorem is referenced by:  ralim  2977  rexim  3037  r19.26  3093  iiner  7862  ss2ixp  7963  undifixp  7986  boxriin  7992  acni2  8907  axcc4  9299  intgru  9674  ingru  9675  prdsdsval3  16192  mrcmndind  17413  hauscmplem  21257  uspgr2wlkeq  26598  wlkp1lem8  26633  prdstotbnd  33723
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