ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elsb4 GIF version

Theorem elsb4 1869
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 ax-17 1435 . . . . 5 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
2 elequ2 1617 . . . . 5 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
31, 2sbieh 1689 . . . 4 ([𝑦 / 𝑤]𝑧𝑤𝑧𝑦)
43sbbii 1664 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑦]𝑧𝑦)
5 ax-17 1435 . . . 4 (𝑧𝑤 → ∀𝑦 𝑧𝑤)
65sbco2h 1854 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑤)
74, 6bitr3i 179 . 2 ([𝑥 / 𝑦]𝑧𝑦 ↔ [𝑥 / 𝑤]𝑧𝑤)
8 equsb1 1684 . . . 4 [𝑥 / 𝑤]𝑤 = 𝑥
9 elequ2 1617 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
109sbimi 1663 . . . 4 ([𝑥 / 𝑤]𝑤 = 𝑥 → [𝑥 / 𝑤](𝑧𝑤𝑧𝑥))
118, 10ax-mp 7 . . 3 [𝑥 / 𝑤](𝑧𝑤𝑧𝑥)
12 sbbi 1849 . . 3 ([𝑥 / 𝑤](𝑧𝑤𝑧𝑥) ↔ ([𝑥 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑥))
1311, 12mpbi 137 . 2 ([𝑥 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑥)
14 ax-17 1435 . . 3 (𝑧𝑥 → ∀𝑤 𝑧𝑥)
1514sbh 1675 . 2 ([𝑥 / 𝑤]𝑧𝑥𝑧𝑥)
167, 13, 153bitri 199 1 ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
Colors of variables: wff set class
Syntax hints:  wb 102  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  peano2  4346
  Copyright terms: Public domain W3C validator