MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsb3 Structured version   Visualization version   GIF version

Theorem equsb3 2416
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 2020. (Revised by Wolf Lammen, 21-Sep-2018.)
Assertion
Ref Expression
equsb3 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Distinct variable group:   𝑦,𝑧

Proof of Theorem equsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 2415 . . 3 ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧)
21sbbii 1873 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3 sbcom3 2395 . . 3 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧)
4 nfv 1829 . . . 4 𝑤[𝑥 / 𝑦]𝑦 = 𝑧
54sbf 2364 . . 3 ([𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
63, 5bitri 262 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
7 equsb3lem 2415 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
82, 6, 73bitr3i 288 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 194  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-13 2229
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by:  sb8eu  2487  mo3  2491  sb8iota  5758  mo5f  28511  mptsnunlem  32161  wl-equsb3  32316  wl-mo3t  32337  wl-sb8eut  32338  frege55lem1b  37009  sbeqal1  37420
  Copyright terms: Public domain W3C validator