Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsb3ALT Structured version   Visualization version   GIF version

Theorem equsb3ALT 2432
 Description: Alternate proof of equsb3 2431, shorter but requiring ax-11 2033. (Contributed by Raph Levien and FL, 4-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
equsb3ALT ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Distinct variable group:   𝑦,𝑧

Proof of Theorem equsb3ALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 2430 . . 3 ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧)
21sbbii 1886 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3 nfv 1842 . . 3 𝑤 𝑦 = 𝑧
43sbco2 2414 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
5 equsb3lem 2430 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
62, 4, 53bitr3i 290 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  [wsb 1879 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator