Theorem ralrimivvva 2616

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 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → 𝜓)

Assertion

Ref	Expression
ralrimivvva	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 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

Step	Hyp	Ref	Expression
1		ralrimivvva.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → 𝜓)
2	1	3anassrs 1256	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜓)
3	2	ralrimiva 2606	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜓)
4	3	ralrimiva 2606	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
5	4	ralrimiva 2606	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   ∈ wcel 2202  ∀wral 2511

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-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-nf 1510  df-ral 2516

This theorem is referenced by:  ispod  4407  swopolem  4408  ordwe  4680  wessep  4682  isopolem  5973  caovassg  6191  caovcang  6194  caovordig  6198  caovordg  6200  caovdig  6207  caovdirg  6210  caoftrn  6277  netap  7533  2omotaplemap  7536  isrngd  14047  isringd  14135  aprap  14382  islmodd  14389  rnglidlmsgrp  14593  rnglidlrng  14594