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Theorem r3al 2534
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 2473 . 2 (∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
2 r2al 2509 . . 3 (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
32ralbii 2496 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
4 3anass 984 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)))
54imbi1i 238 . . . . . . . 8 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑))
6 impexp 263 . . . . . . . 8 (((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
75, 6bitri 184 . . . . . . 7 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
87albii 1481 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
9 19.21v 1884 . . . . . 6 (∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
108, 9bitri 184 . . . . 5 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1110albii 1481 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
12 19.21v 1884 . . . 4 (∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1311, 12bitri 184 . . 3 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1413albii 1481 . 2 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
151, 3, 143bitr4i 212 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wal 1362  wcel 2160  wral 2468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  pocl  4324  soss  4335
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