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Theorem ralimdaa 2403
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
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 . . . . 5 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 112 . . . 4 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
43a2d 26 . . 3 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐴𝜒)))
51, 4alimd 1430 . 2 (𝜑 → (∀𝑥(𝑥𝐴𝜓) → ∀𝑥(𝑥𝐴𝜒)))
6 df-ral 2328 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
7 df-ral 2328 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
85, 6, 73imtr4g 198 1 (𝜑 → (∀𝑥𝐴 𝜓 → ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wnf 1365  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  ralimdva  2404
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