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Theorem ralimdaa 2952
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.)
Hypotheses
Ref Expression
ralimdaa.1 𝑥𝜑
ralimdaa.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralimdaa (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ralimdaa
StepHypRef Expression
1 ralimdaa.1 . . 3 𝑥𝜑
2 ralimdaa.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 450 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 2951 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ralim 2943 . 2 (∀𝑥𝐴 (𝜓𝜒) → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
64, 5syl 17 1 (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wnf 1705  wcel 1987  wral 2907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707  df-ral 2912
This theorem is referenced by:  eltsk2g  9517  ptcnplem  21334  poimirlem26  33067  allbutfifvre  39311  climleltrp  39312  fnlimabslt  39315  stoweidlem61  39585  stoweid  39587  fourierdlem73  39703  smflimlem2  40287
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