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Theorem elsb4 1853
 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 1419 . . . . 5 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
2 elequ2 1601 . . . . 5 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
31, 2sbieh 1673 . . . 4 ([𝑦 / 𝑤]𝑧𝑤𝑧𝑦)
43sbbii 1648 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑦]𝑧𝑦)
5 ax-17 1419 . . . 4 (𝑧𝑤 → ∀𝑦 𝑧𝑤)
65sbco2h 1838 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑤)
74, 6bitr3i 175 . 2 ([𝑥 / 𝑦]𝑧𝑦 ↔ [𝑥 / 𝑤]𝑧𝑤)
8 equsb1 1668 . . . 4 [𝑥 / 𝑤]𝑤 = 𝑥
9 elequ2 1601 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
109sbimi 1647 . . . 4 ([𝑥 / 𝑤]𝑤 = 𝑥 → [𝑥 / 𝑤](𝑧𝑤𝑧𝑥))
118, 10ax-mp 7 . . 3 [𝑥 / 𝑤](𝑧𝑤𝑧𝑥)
12 sbbi 1833 . . 3 ([𝑥 / 𝑤](𝑧𝑤𝑧𝑥) ↔ ([𝑥 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑥))
1311, 12mpbi 133 . 2 ([𝑥 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑥)
14 ax-17 1419 . . 3 (𝑧𝑥 → ∀𝑤 𝑧𝑥)
1514sbh 1659 . 2 ([𝑥 / 𝑤]𝑧𝑥𝑧𝑥)
167, 13, 153bitri 195 1 ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  peano2  4318
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