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

Theorem cbvexd 2401
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2444. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker cbvexdw 2329 if possible. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvald.1 𝑦𝜑
cbvald.2 (𝜑 → Ⅎ𝑦𝜓)
cbvald.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvexd (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvexd
StepHypRef Expression
1 cbvald.1 . . . 4 𝑦𝜑
2 cbvald.2 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
32nfnd 1853 . . . 4 (𝜑 → Ⅎ𝑦 ¬ 𝜓)
4 cbvald.3 . . . . 5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
5 notbi 319 . . . . 5 ((𝜓𝜒) ↔ (¬ 𝜓 ↔ ¬ 𝜒))
64, 5imbitrdi 250 . . . 4 (𝜑 → (𝑥 = 𝑦 → (¬ 𝜓 ↔ ¬ 𝜒)))
71, 3, 6cbvald 2400 . . 3 (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
8 alnex 1775 . . 3 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
9 alnex 1775 . . 3 (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
107, 8, 93bitr3g 313 . 2 (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
1110con4bid 317 1 (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1531  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-11 2146  ax-12 2163  ax-13 2365
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778
This theorem is referenced by:  cbvexdva  2403  dfid3  5570  axrepndlem2  10587  axunnd  10590  axpowndlem2  10592  axpownd  10595  axregndlem2  10597  axinfndlem1  10599  axacndlem4  10604  wl-mo2df  36945  wl-eudf  36947
  Copyright terms: Public domain W3C validator