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Theorem elsb3 2438
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 nfv 1845 . . 3 𝑦 𝑤𝑧
21sbco2 2419 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑤]𝑤𝑧)
3 nfv 1845 . . . 4 𝑤 𝑦𝑧
4 elequ1 1999 . . . 4 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
53, 4sbie 2412 . . 3 ([𝑦 / 𝑤]𝑤𝑧𝑦𝑧)
65sbbii 1889 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑦]𝑦𝑧)
7 nfv 1845 . . 3 𝑤 𝑥𝑧
8 elequ1 1999 . . 3 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
97, 8sbie 2412 . 2 ([𝑥 / 𝑤]𝑤𝑧𝑥𝑧)
102, 6, 93bitr3i 290 1 ([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883
This theorem is referenced by:  cvjust  2621
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