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Theorem cbvex2v 2367
Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2 2436 with a disjoint variable condition, which does not require ax-13 2392. (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 1858 . . . 4 𝑧 ¬ 𝜑
3 cbval2v.2 . . . . 5 𝑤𝜑
43nfn 1858 . . . 4 𝑤 ¬ 𝜑
5 cbval2v.3 . . . . 5 𝑥𝜓
65nfn 1858 . . . 4 𝑥 ¬ 𝜓
7 cbval2v.4 . . . . 5 𝑦𝜓
87nfn 1858 . . . 4 𝑦 ¬ 𝜓
9 cbval2v.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
109notbid 321 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
112, 4, 6, 8, 10cbval2v 2365 . . 3 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑧𝑤 ¬ 𝜓)
1211notbii 323 . 2 (¬ ∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∀𝑧𝑤 ¬ 𝜓)
13 2exnaln 1830 . 2 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
14 2exnaln 1830 . 2 (∃𝑧𝑤𝜓 ↔ ¬ ∀𝑧𝑤 ¬ 𝜓)
1512, 13, 143bitr4i 306 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536  wex 1781  wnf 1785
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  ax-10 2146  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by:  cbvopab  5120  cbvoprab12  7227  bj-cbvex2vv  34146  or2expropbilem2  43482  ichnreuop  43846
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