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Theorem cbvex2v 2346
Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2 2414 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 14-Sep-2003.) (Revised by BJ, 16-Jun-2019.)
Hypotheses
Ref Expression
cbval2v.1 𝑧𝜑
cbval2v.2 𝑤𝜑
cbval2v.3 𝑥𝜓
cbval2v.4 𝑦𝜓
cbval2v.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvex2v (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvex2v
StepHypRef Expression
1 cbval2v.1 . . . . 5 𝑧𝜑
21nfn 1864 . . . 4 𝑧 ¬ 𝜑
3 cbval2v.2 . . . . 5 𝑤𝜑
43nfn 1864 . . . 4 𝑤 ¬ 𝜑
5 cbval2v.3 . . . . 5 𝑥𝜓
65nfn 1864 . . . 4 𝑥 ¬ 𝜓
7 cbval2v.4 . . . . 5 𝑦𝜓
87nfn 1864 . . . 4 𝑦 ¬ 𝜓
9 cbval2v.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
109notbid 318 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
112, 4, 6, 8, 10cbval2v 2344 . . 3 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑧𝑤 ¬ 𝜓)
12 2nexaln 1836 . . 3 (¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
13 2nexaln 1836 . . 3 (¬ ∃𝑧𝑤𝜓 ↔ ∀𝑧𝑤 ¬ 𝜓)
1411, 12, 133bitr4i 303 . 2 (¬ ∃𝑥𝑦𝜑 ↔ ¬ ∃𝑧𝑤𝜓)
1514con4bii 321 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1540  wex 1786  wnf 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-10 2141  ax-11 2158  ax-12 2175
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1787  df-nf 1791
This theorem is referenced by:  cbvopab  5151  cbvoprab12  7358  bj-cbvex2vv  34980  or2expropbilem2  44495  ichnreuop  44893
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