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Theorem clelsb4 2143
 Description: Substitution applied to an atomic wff (class version of elsb4 1853). (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 1421 . . 3 𝑦 𝐴𝑤
21sbco2 1839 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝐴𝑤 ↔ [𝑥 / 𝑤]𝐴𝑤)
3 nfv 1421 . . . 4 𝑤 𝐴𝑦
4 eleq2 2101 . . . 4 (𝑤 = 𝑦 → (𝐴𝑤𝐴𝑦))
53, 4sbie 1674 . . 3 ([𝑦 / 𝑤]𝐴𝑤𝐴𝑦)
65sbbii 1648 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝐴𝑤 ↔ [𝑥 / 𝑦]𝐴𝑦)
7 nfv 1421 . . 3 𝑤 𝐴𝑥
8 eleq2 2101 . . 3 (𝑤 = 𝑥 → (𝐴𝑤𝐴𝑥))
97, 8sbie 1674 . 2 ([𝑥 / 𝑤]𝐴𝑤𝐴𝑥)
102, 6, 93bitr3i 199 1 ([𝑥 / 𝑦]𝐴𝑦𝐴𝑥)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036 This theorem is referenced by:  peano1  4317  peano2  4318
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