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Theorem bj-cbvex4vv 34203
Description: Version of cbvex4v 2438 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbvex4vv.1 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
bj-cbvex4vv.2 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
Assertion
Ref Expression
bj-cbvex4vv (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
Distinct variable groups:   𝑧,𝑤,𝜒   𝑣,𝑢,𝜑   𝑥,𝑦,𝜓   𝑓,𝑔,𝜓   𝑧,𝑓,𝑔,𝑤   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 bj-cbvex4vv
StepHypRef Expression
1 bj-cbvex4vv.1 . . . 4 ((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))
212exbidv 1925 . . 3 ((𝑥 = 𝑣𝑦 = 𝑢) → (∃𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝜓))
32cbvex2vw 2048 . 2 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑧𝑤𝜓)
4 bj-cbvex4vv.2 . . . 4 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
54cbvex2vw 2048 . . 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 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by: (None)
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