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Theorem clelsb3f 2285
 Description: Substitution applied to an atomic wff (class version of elsb3 1951). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
Hypothesis
Ref Expression
clelsb3f.1 𝑥𝐴
Assertion
Ref Expression
clelsb3f ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)

Proof of Theorem clelsb3f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clelsb3f.1 . . . 4 𝑥𝐴
21nfcri 2275 . . 3 𝑥 𝑤𝐴
32sbco2 1938 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑤]𝑤𝐴)
4 nfv 1508 . . . 4 𝑤 𝑥𝐴
5 eleq1w 2200 . . . 4 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
64, 5sbie 1764 . . 3 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
76sbbii 1738 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑥]𝑥𝐴)
8 nfv 1508 . . 3 𝑤 𝑦𝐴
9 eleq1w 2200 . . 3 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
108, 9sbie 1764 . 2 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
113, 7, 103bitr3i 209 1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∈ wcel 1480  [wsb 1735  Ⅎwnfc 2268 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  df-nfc 2270 This theorem is referenced by:  rmo3f  2881
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