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Theorem elsb4 1902
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 1465 . . . . 5 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
2 elequ2 1649 . . . . 5 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
31, 2sbieh 1721 . . . 4 ([𝑦 / 𝑤]𝑧𝑤𝑧𝑦)
43sbbii 1696 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑦]𝑧𝑦)
5 ax-17 1465 . . . 4 (𝑧𝑤 → ∀𝑦 𝑧𝑤)
65sbco2h 1887 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑤)
74, 6bitr3i 185 . 2 ([𝑥 / 𝑦]𝑧𝑦 ↔ [𝑥 / 𝑤]𝑧𝑤)
8 equsb1 1716 . . . 4 [𝑥 / 𝑤]𝑤 = 𝑥
9 elequ2 1649 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
109sbimi 1695 . . . 4 ([𝑥 / 𝑤]𝑤 = 𝑥 → [𝑥 / 𝑤](𝑧𝑤𝑧𝑥))
118, 10ax-mp 7 . . 3 [𝑥 / 𝑤](𝑧𝑤𝑧𝑥)
12 sbbi 1882 . . 3 ([𝑥 / 𝑤](𝑧𝑤𝑧𝑥) ↔ ([𝑥 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑥))
1311, 12mpbi 144 . 2 ([𝑥 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑥)
14 ax-17 1465 . . 3 (𝑧𝑥 → ∀𝑤 𝑧𝑥)
1514sbh 1707 . 2 ([𝑥 / 𝑤]𝑧𝑥𝑧𝑥)
167, 13, 153bitri 205 1 ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694
This theorem is referenced by:  peano2  4423
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