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Theorem cbvex2 2435
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker cbvex2v 2366 if possible. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbval2.1 𝑧𝜑
cbval2.2 𝑤𝜑
cbval2.3 𝑥𝜓
cbval2.4 𝑦𝜓
cbval2.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvex2 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧   𝑥,𝑤   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvex2
StepHypRef Expression
1 cbval2.1 . . . . 5 𝑧𝜑
21nfn 1858 . . . 4 𝑧 ¬ 𝜑
3 cbval2.2 . . . . 5 𝑤𝜑
43nfn 1858 . . . 4 𝑤 ¬ 𝜑
5 cbval2.3 . . . . 5 𝑥𝜓
65nfn 1858 . . . 4 𝑥 ¬ 𝜓
7 cbval2.4 . . . . 5 𝑦𝜓
87nfn 1858 . . . 4 𝑦 ¬ 𝜓
9 cbval2.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
109notbid 321 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
112, 4, 6, 8, 10cbval2 2433 . . 3 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑧𝑤 ¬ 𝜓)
12 2nexaln 1831 . . 3 (¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
13 2nexaln 1831 . . 3 (¬ ∃𝑧𝑤𝜓 ↔ ∀𝑧𝑤 ¬ 𝜓)
1411, 12, 133bitr4i 306 . 2 (¬ ∃𝑥𝑦𝜑 ↔ ¬ ∃𝑧𝑤𝜓)
1514con4bii 324 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 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391
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: (None)
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