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Theorem rr19.28v 3378
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4099 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 472 . . . . . 6 ((𝜑𝜓) → 𝜑)
21ralimi 2981 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜑)
3 biidd 252 . . . . . 6 (𝑦 = 𝑥 → (𝜑𝜑))
43rspcv 3336 . . . . 5 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
52, 4syl5 34 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → 𝜑))
6 simpr 476 . . . . . 6 ((𝜑𝜓) → 𝜓)
76ralimi 2981 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓)
87a1i 11 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓))
95, 8jcad 554 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → (𝜑 ∧ ∀𝑦𝐴 𝜓)))
109ralimia 2979 . 2 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) → ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
11 r19.28v 3100 . . 3 ((𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑦𝐴 (𝜑𝜓))
1211ralimi 2981 . 2 (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
1310, 12impbii 199 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233
This theorem is referenced by: (None)
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