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Theorem clelsb4 2270
Description: Substitution applied to an atomic wff (class version of elsb4 2143). (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 1515 . . 3 𝑥 𝐴𝑤
21sbco2 1952 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴𝑤 ↔ [𝑦 / 𝑤]𝐴𝑤)
3 nfv 1515 . . . 4 𝑤 𝐴𝑥
4 eleq2 2228 . . . 4 (𝑤 = 𝑥 → (𝐴𝑤𝐴𝑥))
53, 4sbie 1778 . . 3 ([𝑥 / 𝑤]𝐴𝑤𝐴𝑥)
65sbbii 1752 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴𝑤 ↔ [𝑦 / 𝑥]𝐴𝑥)
7 nfv 1515 . . 3 𝑤 𝐴𝑦
8 eleq2 2228 . . 3 (𝑤 = 𝑦 → (𝐴𝑤𝐴𝑦))
97, 8sbie 1778 . 2 ([𝑦 / 𝑤]𝐴𝑤𝐴𝑦)
102, 6, 93bitr3i 209 1 ([𝑦 / 𝑥]𝐴𝑥𝐴𝑦)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1749  wcel 2135
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-cleq 2157  df-clel 2160
This theorem is referenced by:  peano1  4566  peano2  4567
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