Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-cbvex2v Structured version   Visualization version   GIF version

Theorem bj-cbvex2v 32401
Description: Version of cbvex2 2279 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbval2v.1 𝑧𝜑
bj-cbval2v.2 𝑤𝜑
bj-cbval2v.3 𝑥𝜓
bj-cbval2v.4 𝑦𝜓
bj-cbval2v.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
bj-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 bj-cbvex2v
StepHypRef Expression
1 bj-cbval2v.1 . . . . 5 𝑧𝜑
21nfn 1781 . . . 4 𝑧 ¬ 𝜑
3 bj-cbval2v.2 . . . . 5 𝑤𝜑
43nfn 1781 . . . 4 𝑤 ¬ 𝜑
5 bj-cbval2v.3 . . . . 5 𝑥𝜓
65nfn 1781 . . . 4 𝑥 ¬ 𝜓
7 bj-cbval2v.4 . . . . 5 𝑦𝜓
87nfn 1781 . . . 4 𝑦 ¬ 𝜓
9 bj-cbval2v.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
109notbid 308 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (¬ 𝜑 ↔ ¬ 𝜓))
112, 4, 6, 8, 10bj-cbval2v 32400 . . 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  wnf 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
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:  bj-cbvex2vv  32403
  Copyright terms: Public domain W3C validator