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Theorem r3al 2413
Description: Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
Assertion
Ref Expression
r3al (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧   𝑧,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem r3al
StepHypRef Expression
1 df-ral 2358 . 2 (∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
2 r2al 2390 . . 3 (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
32ralbii 2377 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
4 3anass 924 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)))
54imbi1i 236 . . . . . . . 8 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑))
6 impexp 259 . . . . . . . 8 (((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
75, 6bitri 182 . . . . . . 7 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
87albii 1400 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
9 19.21v 1796 . . . . . 6 (∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
108, 9bitri 182 . . . . 5 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1110albii 1400 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
12 19.21v 1796 . . . 4 (∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1311, 12bitri 182 . . 3 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1413albii 1400 . 2 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
151, 3, 143bitr4i 210 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920  wal 1283  wcel 1434  wral 2353
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 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358
This theorem is referenced by:  pocl  4086  soss  4097
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