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Theorem elsb3 1852
Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb3 ([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)
Distinct variable group:   𝑦,𝑧

Proof of Theorem elsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-17 1419 . . . . 5 (𝑦𝑧 → ∀𝑤 𝑦𝑧)
2 elequ1 1600 . . . . 5 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
31, 2sbieh 1673 . . . 4 ([𝑦 / 𝑤]𝑤𝑧𝑦𝑧)
43sbbii 1648 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑦]𝑦𝑧)
5 ax-17 1419 . . . 4 (𝑤𝑧 → ∀𝑦 𝑤𝑧)
65sbco2h 1838 . . 3 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑤]𝑤𝑧)
74, 6bitr3i 175 . 2 ([𝑥 / 𝑦]𝑦𝑧 ↔ [𝑥 / 𝑤]𝑤𝑧)
8 equsb1 1668 . . . 4 [𝑥 / 𝑤]𝑤 = 𝑥
9 elequ1 1600 . . . . 5 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
109sbimi 1647 . . . 4 ([𝑥 / 𝑤]𝑤 = 𝑥 → [𝑥 / 𝑤](𝑤𝑧𝑥𝑧))
118, 10ax-mp 7 . . 3 [𝑥 / 𝑤](𝑤𝑧𝑥𝑧)
12 sbbi 1833 . . 3 ([𝑥 / 𝑤](𝑤𝑧𝑥𝑧) ↔ ([𝑥 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑤]𝑥𝑧))
1311, 12mpbi 133 . 2 ([𝑥 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑤]𝑥𝑧)
14 ax-17 1419 . . 3 (𝑥𝑧 → ∀𝑤 𝑥𝑧)
1514sbh 1659 . 2 ([𝑥 / 𝑤]𝑥𝑧𝑥𝑧)
167, 13, 153bitri 195 1 ([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)
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-13 1404  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:  cvjust  2035
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