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Theorem r2alf 2341
 Description: Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
r2alf.1 𝑦𝐴
Assertion
Ref Expression
r2alf (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r2alf
StepHypRef Expression
1 df-ral 2311 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 r2alf.1 . . . . . 6 𝑦𝐴
32nfcri 2172 . . . . 5 𝑦 𝑥𝐴
4319.21 1475 . . . 4 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
5 impexp 250 . . . . 5 (((𝑥𝐴𝑦𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑦𝐵𝜑)))
65albii 1359 . . . 4 (∀𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)))
7 df-ral 2311 . . . . 5 (∀𝑦𝐵 𝜑 ↔ ∀𝑦(𝑦𝐵𝜑))
87imbi2i 215 . . . 4 ((𝑥𝐴 → ∀𝑦𝐵 𝜑) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
94, 6, 83bitr4i 201 . . 3 (∀𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ (𝑥𝐴 → ∀𝑦𝐵 𝜑))
109albii 1359 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝜑))
111, 10bitr4i 176 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  Ⅎwnfc 2165  ∀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-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311 This theorem is referenced by:  r2al  2343  ralcomf  2471
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