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Theorem clelsb4 2245
Description: Substitution applied to an atomic wff (class version of elsb4 1952). (Contributed by Jim Kingdon, 22-Nov-2018.)
Assertion
Ref Expression
clelsb4 ([𝑦 / 𝑥]𝐴𝑥𝐴𝑦)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1508 . . 3 𝑥 𝐴𝑤
21sbco2 1938 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴𝑤 ↔ [𝑦 / 𝑤]𝐴𝑤)
3 nfv 1508 . . . 4 𝑤 𝐴𝑥
4 eleq2 2203 . . . 4 (𝑤 = 𝑥 → (𝐴𝑤𝐴𝑥))
53, 4sbie 1764 . . 3 ([𝑥 / 𝑤]𝐴𝑤𝐴𝑥)
65sbbii 1738 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴𝑤 ↔ [𝑦 / 𝑥]𝐴𝑥)
7 nfv 1508 . . 3 𝑤 𝐴𝑦
8 eleq2 2203 . . 3 (𝑤 = 𝑦 → (𝐴𝑤𝐴𝑦))
97, 8sbie 1764 . 2 ([𝑦 / 𝑤]𝐴𝑤𝐴𝑦)
102, 6, 93bitr3i 209 1 ([𝑦 / 𝑥]𝐴𝑥𝐴𝑦)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1480  [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135
This theorem is referenced by:  peano1  4508  peano2  4509
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