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Theorem elsb4 2530
Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Remove dependency on ax-11 2198. (Revised by Wolf Lammen, 28-Jul-2022.)
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 sbcom3 2502 . . 3 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑧𝑦 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑧𝑦)
2 elsb4lem 2529 . . . 4 ([𝑤 / 𝑦]𝑧𝑦𝑧𝑤)
32sbbii 2069 . . 3 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑧𝑦 ↔ [𝑥 / 𝑤]𝑧𝑤)
4 nfv 2009 . . . 4 𝑤[𝑥 / 𝑦]𝑧𝑦
54sbf 2471 . . 3 ([𝑥 / 𝑤][𝑥 / 𝑦]𝑧𝑦 ↔ [𝑥 / 𝑦]𝑧𝑦)
61, 3, 53bitr3i 292 . 2 ([𝑥 / 𝑤]𝑧𝑤 ↔ [𝑥 / 𝑦]𝑧𝑦)
7 elsb4lem 2529 . 2 ([𝑥 / 𝑤]𝑧𝑤𝑧𝑥)
86, 7bitr3i 268 1 ([𝑥 / 𝑦]𝑧𝑦𝑧𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 197  [wsb 2062
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 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879  df-sb 2063
This theorem is referenced by:  nfnid  5013
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