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Theorem rr19.28v 2683
 Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (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 102 . . . . . 6 ((𝜑𝜓) → 𝜑)
21ralimi 2384 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜑)
3 biidd 161 . . . . . 6 (𝑦 = 𝑥 → (𝜑𝜑))
43rspcv 2652 . . . . 5 (𝑥𝐴 → (∀𝑦𝐴 𝜑𝜑))
52, 4syl5 28 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → 𝜑))
6 simpr 103 . . . . . 6 ((𝜑𝜓) → 𝜓)
76ralimi 2384 . . . . 5 (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓)
87a1i 9 . . . 4 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → ∀𝑦𝐴 𝜓))
95, 8jcad 291 . . 3 (𝑥𝐴 → (∀𝑦𝐴 (𝜑𝜓) → (𝜑 ∧ ∀𝑦𝐴 𝜓)))
109ralimia 2382 . 2 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) → ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
11 r19.28av 2449 . . 3 ((𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑦𝐴 (𝜑𝜓))
1211ralimi 2384 . 2 (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) → ∀𝑥𝐴𝑦𝐴 (𝜑𝜓))
1310, 12impbii 117 1 (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559 This theorem is referenced by: (None)
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