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Theorem r3al 2969
Description: Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.) (Proof shortened by Wolf Lammen, 30-Dec-2019.)
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 r2al 2968 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑))
2 19.21v 1908 . . . 4 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧𝐶𝜑)))
3 df-3an 1056 . . . . . . 7 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶))
43imbi1i 338 . . . . . 6 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑))
5 impexp 461 . . . . . 6 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)))
64, 5bitri 264 . . . . 5 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)))
76albii 1787 . . . 4 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)))
8 df-ral 2946 . . . . 5 (∀𝑧𝐶 𝜑 ↔ ∀𝑧(𝑧𝐶𝜑))
98imbi2i 325 . . . 4 (((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧𝐶𝜑)))
102, 7, 93bitr4ri 293 . . 3 (((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑) ↔ ∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
11102albii 1788 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
121, 11bitri 264 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1056  df-ex 1745  df-ral 2946
This theorem is referenced by:  pocl  5071  dfwe2  7023  isass  33775  ntrneikb  38709  ntrneixb  38710
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