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Theorem r3al 2514
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 2453 . 2 (∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
2 r2al 2489 . . 3 (∀𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
32ralbii 2476 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝐴𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑))
4 3anass 977 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)))
54imbi1i 237 . . . . . . . 8 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑))
6 impexp 261 . . . . . . . 8 (((𝑥𝐴 ∧ (𝑦𝐵𝑧𝐶)) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
75, 6bitri 183 . . . . . . 7 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
87albii 1463 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)))
9 19.21v 1866 . . . . . 6 (∀𝑧(𝑥𝐴 → ((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
108, 9bitri 183 . . . . 5 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1110albii 1463 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
12 19.21v 1866 . . . 4 (∀𝑦(𝑥𝐴 → ∀𝑧((𝑦𝐵𝑧𝐶) → 𝜑)) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1311, 12bitri 183 . . 3 (∀𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
1413albii 1463 . 2 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝑧((𝑦𝐵𝑧𝐶) → 𝜑)))
151, 3, 143bitr4i 211 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973  wal 1346  wcel 2141  wral 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453
This theorem is referenced by:  pocl  4286  soss  4297
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