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Theorem cbvral6vw 3241
Description: Change bound variables of sextuple restricted universal quantification, using implicit substitution. (Contributed by Scott Fenton, 5-Mar-2025.)
Hypotheses
Ref Expression
cbvral6vw.1 (𝑥 = 𝑎 → (𝜑𝜒))
cbvral6vw.2 (𝑦 = 𝑏 → (𝜒𝜃))
cbvral6vw.3 (𝑧 = 𝑐 → (𝜃𝜏))
cbvral6vw.4 (𝑤 = 𝑑 → (𝜏𝜂))
cbvral6vw.5 (𝑝 = 𝑒 → (𝜂𝜁))
cbvral6vw.6 (𝑞 = 𝑓 → (𝜁𝜓))
Assertion
Ref Expression
cbvral6vw (∀𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷𝑝𝐸𝑞𝐹 𝜑 ↔ ∀𝑎𝐴𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸𝑓𝐹 𝜓)
Distinct variable groups:   𝑥,𝑎,𝐴   𝑥,𝑦,𝐵,𝑎   𝑦,𝑏,𝐵   𝑥,𝐶,𝑦,𝑎   𝐶,𝑏   𝑧,𝑐,𝐶   𝑥,𝑧,𝐷,𝑦,𝑎   𝐷,𝑏   𝐷,𝑐   𝑤,𝑑,𝐷   𝑥,𝐸,𝑦,𝑧,𝑎   𝐸,𝑏   𝐸,𝑐   𝐸,𝑑   𝑒,𝐸   𝑤,𝑝,𝐸   𝑥,𝐹,𝑦,𝑧,𝑎   𝐹,𝑏   𝐹,𝑐   𝐹,𝑑   𝑒,𝐹   𝑓,𝑞,𝐹   𝐹,𝑝,𝑤,𝑎   𝜑,𝑎   𝑞,𝑎,𝑤   𝜒,𝑏   𝑝,𝑏,𝑞,𝑧,𝑤   𝑝,𝑐,𝑞   𝜃,𝑐   𝑤,𝑐   𝜒,𝑥   𝑝,𝑑,𝑞   𝜏,𝑑   𝜂,𝑒   𝑒,𝑝,𝑞   𝜂,𝑤   𝜁,𝑓   𝑥,𝑞,𝑦,𝑝   𝜁,𝑝   𝜓,𝑞   𝑥,𝑤,𝑦   𝜏,𝑧   𝜃,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑏,𝑐,𝑑)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑒,𝑓,𝑝,𝑎,𝑏,𝑐,𝑑)   𝜒(𝑦,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑎,𝑐,𝑑)   𝜃(𝑥,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑎,𝑏,𝑑)   𝜏(𝑥,𝑦,𝑤,𝑒,𝑓,𝑞,𝑝,𝑎,𝑏,𝑐)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝜁(𝑥,𝑦,𝑧,𝑤,𝑒,𝑞,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑦,𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑏,𝑐,𝑑)   𝐵(𝑧,𝑤,𝑒,𝑓,𝑞,𝑝,𝑐,𝑑)   𝐶(𝑤,𝑒,𝑓,𝑞,𝑝,𝑑)   𝐷(𝑒,𝑓,𝑞,𝑝)   𝐸(𝑓,𝑞)

Proof of Theorem cbvral6vw
StepHypRef Expression
1 cbvral6vw.1 . . . 4 (𝑥 = 𝑎 → (𝜑𝜒))
212ralbidv 3217 . . 3 (𝑥 = 𝑎 → (∀𝑝𝐸𝑞𝐹 𝜑 ↔ ∀𝑝𝐸𝑞𝐹 𝜒))
3 cbvral6vw.2 . . . 4 (𝑦 = 𝑏 → (𝜒𝜃))
432ralbidv 3217 . . 3 (𝑦 = 𝑏 → (∀𝑝𝐸𝑞𝐹 𝜒 ↔ ∀𝑝𝐸𝑞𝐹 𝜃))
5 cbvral6vw.3 . . . 4 (𝑧 = 𝑐 → (𝜃𝜏))
652ralbidv 3217 . . 3 (𝑧 = 𝑐 → (∀𝑝𝐸𝑞𝐹 𝜃 ↔ ∀𝑝𝐸𝑞𝐹 𝜏))
7 cbvral6vw.4 . . . 4 (𝑤 = 𝑑 → (𝜏𝜂))
872ralbidv 3217 . . 3 (𝑤 = 𝑑 → (∀𝑝𝐸𝑞𝐹 𝜏 ↔ ∀𝑝𝐸𝑞𝐹 𝜂))
92, 4, 6, 8cbvral4vw 3240 . 2 (∀𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷𝑝𝐸𝑞𝐹 𝜑 ↔ ∀𝑎𝐴𝑏𝐵𝑐𝐶𝑑𝐷𝑝𝐸𝑞𝐹 𝜂)
10 cbvral6vw.5 . . . 4 (𝑝 = 𝑒 → (𝜂𝜁))
11 cbvral6vw.6 . . . 4 (𝑞 = 𝑓 → (𝜁𝜓))
1210, 11cbvral2vw 3237 . . 3 (∀𝑝𝐸𝑞𝐹 𝜂 ↔ ∀𝑒𝐸𝑓𝐹 𝜓)
13124ralbii 3130 . 2 (∀𝑎𝐴𝑏𝐵𝑐𝐶𝑑𝐷𝑝𝐸𝑞𝐹 𝜂 ↔ ∀𝑎𝐴𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸𝑓𝐹 𝜓)
149, 13bitri 274 1 (∀𝑥𝐴𝑦𝐵𝑧𝐶𝑤𝐷𝑝𝐸𝑞𝐹 𝜑 ↔ ∀𝑎𝐴𝑏𝐵𝑐𝐶𝑑𝐷𝑒𝐸𝑓𝐹 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wral 3060
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 1913  ax-6 1971  ax-7 2011  ax-8 2108
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-clel 2809  df-ral 3061
This theorem is referenced by:  mulsproplemcbv  27481  mulsprop  27496
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