MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsb4 Structured version   Visualization version   GIF version

Theorem elsb4 2434
Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb4 ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
Distinct variable group:   𝑦,𝑧

Proof of Theorem elsb4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1842 . . 3 𝑦 𝑧𝑤
21sbco2 2414 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑤)
3 nfv 1842 . . . 4 𝑤 𝑧𝑦
4 elequ2 2003 . . . 4 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
53, 4sbie 2407 . . 3 ([𝑦 / 𝑤]𝑧𝑤𝑧𝑦)
65sbbii 1886 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑦]𝑧𝑦)
7 nfv 1842 . . 3 𝑤 𝑧𝑥
8 elequ2 2003 . . 3 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
97, 8sbie 2407 . 2 ([𝑥 / 𝑤]𝑧𝑤𝑧𝑥)
102, 6, 93bitr3i 290 1 ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
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-9 1998  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:  nfnid  4895
  Copyright terms: Public domain W3C validator