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Theorem ralrimivvva 2450
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
ralrimivvva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵𝑧𝐶)) → 𝜓)
Assertion
Ref Expression
ralrimivvva (𝜑 → ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜓)
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑦,𝐴,𝑧   𝑧,𝐵
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem ralrimivvva
StepHypRef Expression
1 ralrimivvva.1 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐵𝑧𝐶)) → 𝜓)
213anassrs 1161 . . . 4 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝑧𝐶) → 𝜓)
32ralrimiva 2440 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → ∀𝑧𝐶 𝜓)
43ralrimiva 2440 . 2 ((𝜑𝑥𝐴) → ∀𝑦𝐵𝑧𝐶 𝜓)
54ralrimiva 2440 1 (𝜑 → ∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920  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-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-ral 2358
This theorem is referenced by:  ispod  4095  swopolem  4096  ordwe  4354  wessep  4356  isopolem  5540  caovassg  5738  caovcang  5741  caovordig  5745  caovordg  5747  caovdig  5754  caovdirg  5757  caoftrn  5815
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