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Theorem cbvex4v 2437
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbvex4vw 2049 if possible. (Contributed by NM, 26-Jul-1995.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvex4v.1 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
cbvex4v.2 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
Assertion
Ref Expression
cbvex4v (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
Distinct variable groups:   𝑧,𝑤,𝜒   𝑣,𝑢,𝜑   𝑥,𝑦,𝜓   𝑓,𝑔,𝜓   𝑤,𝑓   𝑧,𝑔   𝑤,𝑢,𝑧,𝑣   𝑥,𝑢,𝑤,𝑧   𝑦,𝑣,𝑤,𝑧
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 cbvex4v
StepHypRef Expression
1 cbvex4v.1 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
212exbidv 1925 . . 3 ((𝑥 = 𝑣𝑦 = 𝑢) → (∃𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝜓))
32cbvex2vv 2436 . 2 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑧𝑤𝜓)
4 cbvex4v.2 . . . 4 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
54cbvex2vv 2436 . . 3 (∃𝑧𝑤𝜓 ↔ ∃𝑓𝑔𝜒)
652exbii 1849 . 2 (∃𝑣𝑢𝑧𝑤𝜓 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
73, 6bitri 277 1 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wex 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2161  ax-12 2177  ax-13 2390
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785
This theorem is referenced by: (None)
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