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

Theorem bj-cbvexw 33001
 Description: Change bound variable. This is to cbvexvw 2126 what cbvalw 2124 is to cbvalvw 2125. (Contributed by BJ, 17-Mar-2020.)
Hypotheses
Ref Expression
bj-cbvexw.1 (∃𝑥𝑦𝜓 → ∃𝑦𝜓)
bj-cbvexw.2 (𝜑 → ∀𝑦𝜑)
bj-cbvexw.3 (∃𝑦𝑥𝜑 → ∃𝑥𝜑)
bj-cbvexw.4 (𝜓 → ∀𝑥𝜓)
bj-cbvexw.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-cbvexw (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-cbvexw
StepHypRef Expression
1 bj-cbvexw.1 . . 3 (∃𝑥𝑦𝜓 → ∃𝑦𝜓)
2 bj-cbvexw.2 . . 3 (𝜑 → ∀𝑦𝜑)
3 bj-cbvexw.5 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
43equcoms 2105 . . . 4 (𝑦 = 𝑥 → (𝜑𝜓))
54biimpd 219 . . 3 (𝑦 = 𝑥 → (𝜑𝜓))
61, 2, 5bj-cbvexiw 32996 . 2 (∃𝑥𝜑 → ∃𝑦𝜓)
7 bj-cbvexw.3 . . 3 (∃𝑦𝑥𝜑 → ∃𝑥𝜑)
8 bj-cbvexw.4 . . 3 (𝜓 → ∀𝑥𝜓)
93biimprd 238 . . 3 (𝑥 = 𝑦 → (𝜓𝜑))
107, 8, 9bj-cbvexiw 32996 . 2 (∃𝑦𝜓 → ∃𝑥𝜑)
116, 10impbii 199 1 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1629  ∃wex 1852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator