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Theorem rr19.28v 3666
 Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4449 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 483 . . . . . 6 ((𝜑𝜓) → 𝜑)
21ralimi 3165 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜑)
3 biidd 263 . . . . . 6 (𝑦 = 𝑥 → (𝜑𝜑))
43rspcv 3622 . . . . 5 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
52, 4syl5 34 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → 𝜑))
6 simpr 485 . . . . 5 ((𝜑𝜓) → 𝜓)
76ralimi 3165 . . . 4 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓)
85, 7jca2 514 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → (𝜑 ∧ ∀𝑦𝐴 𝜓)))
98ralimia 3163 . 2 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) → ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
10 r19.28v 3191 . . 3 ((𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑦𝐴 (𝜑𝜓))
1110ralimi 3165 . 2 (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
129, 11impbii 210 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   ∈ wcel 2107  ∀wral 3143 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2819  df-clel 2898  df-ral 3148 This theorem is referenced by: (None)
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