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Theorem cbvex4vw 2050
Description: Rule used to change bound variables, using implicit substitution. Version of cbvex4v 2439 with more disjoint variable conditions, which requires fewer axioms. (Contributed by NM, 26-Jul-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvex4vw.1 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
cbvex4vw.2 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
Assertion
Ref Expression
cbvex4vw (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
Distinct variable groups:   𝑧,𝑤,𝜒   𝑣,𝑢,𝜑   𝑥,𝑦,𝜓   𝑓,𝑔,𝜓   𝑤,𝑓,𝑧,𝑔   𝑤,𝑢,𝑥,𝑦,𝑧,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)   𝜓(𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)

Proof of Theorem cbvex4vw
StepHypRef Expression
1 cbvex4vw.1 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
212exbidv 1926 . . 3 ((𝑥 = 𝑣𝑦 = 𝑢) → (∃𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝜓))
32cbvex2vw 2049 . 2 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑧𝑤𝜓)
4 cbvex4vw.2 . . . 4 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
54cbvex2vw 2049 . . 3 (∃𝑧𝑤𝜓 ↔ ∃𝑓𝑔𝜒)
652exbii 1850 . 2 (∃𝑣𝑢𝑧𝑤𝜓 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
73, 6bitri 278 1 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781
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 1912  ax-6 1971  ax-7 2016
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  addsrmo  10482  mulsrmo  10483
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