Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvex2 Structured version   Visualization version   GIF version

Theorem cbvex2 2279
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.)
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 1781 . . . 4 𝑧 ¬ 𝜑
3 cbval2.2 . . . . 5 𝑤𝜑
43nfn 1781 . . . 4 𝑤 ¬ 𝜑
5 cbval2.3 . . . . 5 𝑥𝜓
65nfn 1781 . . . 4 𝑥 ¬ 𝜓
7 cbval2.4 . . . . 5 𝑦𝜓
87nfn 1781 . . . 4 𝑦 ¬ 𝜓
9 cbval2.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
109notbid 308 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
112, 4, 6, 8, 10cbval2 2278 . . 3 (∀𝑥𝑦 ¬ 𝜑 ↔ ∀𝑧𝑤 ¬ 𝜓)
1211notbii 310 . 2 (¬ ∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∀𝑧𝑤 ¬ 𝜓)
13 2exnaln 1753 . 2 (∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
14 2exnaln 1753 . 2 (∃𝑧𝑤𝜓 ↔ ¬ ∀𝑧𝑤 ¬ 𝜓)
1512, 13, 143bitr4i 292 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478  ∃wex 1701  Ⅎwnf 1705 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707 This theorem is referenced by:  cbvex2vOLD  2287  cbvopab  4693  cbvoprab12  6694
 Copyright terms: Public domain W3C validator