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Theorem clelsb3 2611
Description: Substitution applied to an atomic wff (class version of elsb3 2317). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb3 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem clelsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1792 . . 3 𝑦 𝑤𝐴
21sbco2 2298 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
3 nfv 1792 . . . 4 𝑤 𝑦𝐴
4 eleq1 2571 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
53, 4sbie 2291 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
65sbbii 1835 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
7 nfv 1792 . . 3 𝑤 𝑥𝐴
8 eleq1 2571 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
97, 8sbie 2291 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
102, 6, 93bitr3i 285 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 191  [wsb 1828  wcel 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829  df-cleq 2498  df-clel 2501
This theorem is referenced by:  hblem  2613  cbvreu  3038  sbcel1v  3350  rmo3  3380  kmlem15  8679  iuninc  28335  measiuns  29194  ballotlemodife  29484  bj-nfcf  31711  sbcel1gvOLD  37603  ellimcabssub0  38101
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