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Theorem 3rspcedvdw 39944
Description: Triple application of rspcedvdw 39942. (Contributed by SN, 20-Aug-2024.)
Hypotheses
Ref Expression
3rspcedvdw.1 (𝑥 = 𝐴 → (𝜓𝜒))
3rspcedvdw.2 (𝑦 = 𝐵 → (𝜒𝜃))
3rspcedvdw.3 (𝑧 = 𝐶 → (𝜃𝜏))
3rspcedvdw.a (𝜑𝐴𝑋)
3rspcedvdw.b (𝜑𝐵𝑌)
3rspcedvdw.c (𝜑𝐶𝑍)
3rspcedvdw.4 (𝜑𝜏)
Assertion
Ref Expression
3rspcedvdw (𝜑 → ∃𝑥𝑋𝑦𝑌𝑧𝑍 𝜓)
Distinct variable groups:   𝜒,𝑥   𝜃,𝑦   𝜏,𝑧   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑋   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑥,𝑧)   𝜏(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝑋(𝑦,𝑧)   𝑌(𝑧)

Proof of Theorem 3rspcedvdw
StepHypRef Expression
1 3rspcedvdw.a . 2 (𝜑𝐴𝑋)
2 3rspcedvdw.b . 2 (𝜑𝐵𝑌)
3 3rspcedvdw.c . 2 (𝜑𝐶𝑍)
4 3rspcedvdw.4 . 2 (𝜑𝜏)
5 3rspcedvdw.1 . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
6 3rspcedvdw.2 . . 3 (𝑦 = 𝐵 → (𝜒𝜃))
7 3rspcedvdw.3 . . 3 (𝑧 = 𝐶 → (𝜃𝜏))
85, 6, 7rspc3ev 3566 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝜏) → ∃𝑥𝑋𝑦𝑌𝑧𝑍 𝜓)
91, 2, 3, 4, 8syl31anc 1375 1 (𝜑 → ∃𝑥𝑋𝑦𝑌𝑧𝑍 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2112  wrex 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070
This theorem is referenced by:  dffltz  40221  nna4b4nsq  40247
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