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Theorem r2alf 2384
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 2354 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 r2alf.1 . . . . . 6 𝑦𝐴
32nfcri 2214 . . . . 5 𝑦 𝑥𝐴
4319.21 1516 . . . 4 (∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
5 impexp 259 . . . . 5 (((𝑥𝐴𝑦𝐵) → 𝜑) ↔ (𝑥𝐴 → (𝑦𝐵𝜑)))
65albii 1400 . . . 4 (∀𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑦(𝑥𝐴 → (𝑦𝐵𝜑)))
7 df-ral 2354 . . . . 5 (∀𝑦𝐵 𝜑 ↔ ∀𝑦(𝑦𝐵𝜑))
87imbi2i 224 . . . 4 ((𝑥𝐴 → ∀𝑦𝐵 𝜑) ↔ (𝑥𝐴 → ∀𝑦(𝑦𝐵𝜑)))
94, 6, 83bitr4i 210 . . 3 (∀𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ (𝑥𝐴 → ∀𝑦𝐵 𝜑))
109albii 1400 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝜑))
111, 10bitr4i 185 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wcel 1434  wnfc 2207  wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354
This theorem is referenced by:  r2al  2386  ralcomf  2516
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