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Theorem r3al 3170
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 3169 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑))
2 19.21v 1940 . . . 4 (∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧𝐶𝜑)))
3 df-3an 1086 . . . . . . 7 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶))
43imbi1i 353 . . . . . 6 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑))
5 impexp 454 . . . . . 6 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)))
64, 5bitri 278 . . . . 5 (((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)))
76albii 1821 . . . 4 (∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑) ↔ ∀𝑧((𝑥𝐴𝑦𝐵) → (𝑧𝐶𝜑)))
8 df-ral 3114 . . . . 5 (∀𝑧𝐶 𝜑 ↔ ∀𝑧(𝑧𝐶𝜑))
98imbi2i 339 . . . 4 (((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑) ↔ ((𝑥𝐴𝑦𝐵) → ∀𝑧(𝑧𝐶𝜑)))
102, 7, 93bitr4ri 307 . . 3 (((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑) ↔ ∀𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
11102albii 1822 . 2 (∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
121, 11bitri 278 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536  wcel 2112  wral 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-ral 3114
This theorem is referenced by:  pocl  5449  dfwe2  7480  isass  35283  dfeldisj3  36111
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