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

Theorem cbvmo 2689
 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 cbvmow 2687 when possible. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 4-Jan-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvmo.1 𝑦𝜑
cbvmo.2 𝑥𝜓
cbvmo.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvmo (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Proof of Theorem cbvmo
StepHypRef Expression
1 cbvmo.1 . . 3 𝑦𝜑
21sb8mo 2686 . 2 (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
3 cbvmo.2 . . . 4 𝑥𝜓
4 cbvmo.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 2544 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65mobii 2630 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓)
72, 6bitri 278 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1785  [wsb 2069  ∃*wmo 2620 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-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653 This theorem is referenced by:  cbveuALT  2693
 Copyright terms: Public domain W3C validator