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Theorem rr19.28v 3536
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4259 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)
Assertion
Ref Expression
rr19.28v (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rr19.28v
StepHypRef Expression
1 simpl 475 . . . . . 6 ((𝜑𝜓) → 𝜑)
21ralimi 3133 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜑)
3 biidd 254 . . . . . 6 (𝑦 = 𝑥 → (𝜑𝜑))
43rspcv 3493 . . . . 5 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
52, 4syl5 34 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → 𝜑))
6 simpr 478 . . . . . 6 ((𝜑𝜓) → 𝜓)
76ralimi 3133 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓)
87a1i 11 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓))
95, 8jcad 509 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → (𝜑 ∧ ∀𝑦𝐴 𝜓)))
109ralimia 3131 . 2 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) → ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
11 r19.28v 3252 . . 3 ((𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑦𝐴 (𝜑𝜓))
1211ralimi 3133 . 2 (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
1310, 12impbii 201 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wral 3089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387
This theorem is referenced by: (None)
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