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Theorem elsb3OLD 2528
Description: Obsolete version of elsb3 2527 as of 27-Jul-2022. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elsb3OLD ([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)
Distinct variable group:   𝑦,𝑧

Proof of Theorem elsb3OLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 2009 . . 3 𝑦 𝑤𝑧
21sbco2 2506 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑤]𝑤𝑧)
3 nfv 2009 . . . 4 𝑤 𝑦𝑧
4 elequ1 2162 . . . 4 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
53, 4sbie 2499 . . 3 ([𝑦 / 𝑤]𝑤𝑧𝑦𝑧)
65sbbii 2068 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝑧 ↔ [𝑥 / 𝑦]𝑦𝑧)
7 nfv 2009 . . 3 𝑤 𝑥𝑧
8 elequ1 2162 . . 3 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
97, 8sbie 2499 . 2 ([𝑥 / 𝑤]𝑤𝑧𝑥𝑧)
102, 6, 93bitr3i 292 1 ([𝑥 / 𝑦]𝑦𝑧𝑥𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 197  [wsb 2061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062
This theorem is referenced by: (None)
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