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Theorem clelsb3fOLD 2937
 Description: Obsolete version of clelsb3f 2936 as of 7-May-2023. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
clelsb3f.1 𝑦𝐴
Assertion
Ref Expression
clelsb3fOLD ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)

Proof of Theorem clelsb3fOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 clelsb3f.1 . . . 4 𝑦𝐴
21nfcri 2926 . . 3 𝑦 𝑤𝐴
32sbco2 2477 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴)
4 nfv 1873 . . . 4 𝑤 𝑦𝐴
5 eleq1w 2848 . . . 4 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
64, 5sbie 2468 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
76sbbii 2027 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
8 nfv 1873 . . 3 𝑤 𝑥𝐴
9 eleq1w 2848 . . 3 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
108, 9sbie 2468 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
113, 7, 103bitr3i 293 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  [wsb 2015   ∈ wcel 2050  Ⅎwnfc 2916 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clel 2846  df-nfc 2918 This theorem is referenced by: (None)
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