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Theorem clelsb4 2185
Description: Substitution applied to an atomic wff (class version of elsb4 1895). (Contributed by Jim Kingdon, 22-Nov-2018.)
Assertion
Ref Expression
clelsb4 ([𝑥 / 𝑦]𝐴𝑦𝐴𝑥)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem clelsb4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . 3 𝑦 𝐴𝑤
21sbco2 1881 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝐴𝑤 ↔ [𝑥 / 𝑤]𝐴𝑤)
3 nfv 1462 . . . 4 𝑤 𝐴𝑦
4 eleq2 2143 . . . 4 (𝑤 = 𝑦 → (𝐴𝑤𝐴𝑦))
53, 4sbie 1715 . . 3 ([𝑦 / 𝑤]𝐴𝑤𝐴𝑦)
65sbbii 1689 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝐴𝑤 ↔ [𝑥 / 𝑦]𝐴𝑦)
7 nfv 1462 . . 3 𝑤 𝐴𝑥
8 eleq2 2143 . . 3 (𝑤 = 𝑥 → (𝐴𝑤𝐴𝑥))
97, 8sbie 1715 . 2 ([𝑥 / 𝑤]𝐴𝑤𝐴𝑥)
102, 6, 93bitr3i 208 1 ([𝑥 / 𝑦]𝐴𝑦𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1434  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078
This theorem is referenced by:  peano1  4343  peano2  4344
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