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Theorem bj-cbvex4vv 33079
Description: Version of cbvex4v 2444 with a dv condition, which does not require ax-13 2408. (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 2004 . . 3 ((𝑥 = 𝑣𝑦 = 𝑢) → (∃𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝜓))
32bj-cbvex2vv 33076 . 2 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑧𝑤𝜓)
4 bj-cbvex4vv.2 . . . 4 ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))
54bj-cbvex2vv 33076 . . 3 (∃𝑧𝑤𝜓 ↔ ∃𝑓𝑔𝜒)
652exbii 1925 . 2 (∃𝑣𝑢𝑧𝑤𝜓 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
73, 6bitri 264 1 (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858
This theorem is referenced by: (None)
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