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Theorem elsb4OLD 2531
Description: Obsolete version of elsb3 2527 as of 27-Jul-2022. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elsb4OLD ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
Distinct variable group:   𝑦,𝑧

Proof of Theorem elsb4OLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 2009 . . 3 𝑦 𝑧𝑤
21sbco2 2506 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑤]𝑧𝑤)
3 nfv 2009 . . . 4 𝑤 𝑧𝑦
4 elequ2 2169 . . . 4 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
53, 4sbie 2499 . . 3 ([𝑦 / 𝑤]𝑧𝑤𝑧𝑦)
65sbbii 2068 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑦]𝑧𝑦)
7 nfv 2009 . . 3 𝑤 𝑧𝑥
8 elequ2 2169 . . 3 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
97, 8sbie 2499 . 2 ([𝑥 / 𝑤]𝑧𝑤𝑧𝑥)
102, 6, 93bitr3i 292 1 ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
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-9 2164  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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