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Theorem rr19.28v 3314
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the nonempty class condition of r19.28zv 4017 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 471 . . . . . 6 ((𝜑𝜓) → 𝜑)
21ralimi 2935 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜑)
3 biidd 250 . . . . . 6 (𝑦 = 𝑥 → (𝜑𝜑))
43rspcv 3277 . . . . 5 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
52, 4syl5 33 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → 𝜑))
6 simpr 475 . . . . . 6 ((𝜑𝜓) → 𝜓)
76ralimi 2935 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓)
87a1i 11 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓))
95, 8jcad 553 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → (𝜑 ∧ ∀𝑦𝐴 𝜓)))
109ralimia 2933 . 2 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) → ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
11 r19.28v 3052 . . 3 ((𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑦𝐴 (𝜑𝜓))
1211ralimi 2935 . 2 (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
1310, 12impbii 197 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  wral 2895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-v 3174
This theorem is referenced by: (None)
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