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Theorem ral2imi 2439
Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
Hypothesis
Ref Expression
ral2imi.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ral2imi (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ral2imi
StepHypRef Expression
1 ral2imi.1 . . 3 (𝜑 → (𝜓𝜒))
21ralimi 2438 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 ralim 2434 . 2 (∀𝑥𝐴 (𝜓𝜒) → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
42, 3syl 14 1 (∀𝑥𝐴 𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383
This theorem depends on definitions:  df-bi 115  df-ral 2364
This theorem is referenced by:  r19.26  2497  iinerm  6364  bj-findis  11874
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