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Theorem clelsb3vOLD 2945
 Description: Obsolete version of clelsb3 2944 as of 29-Jul-2023. (Contributed by Wolf Lammen, 30-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
clelsb3vOLD ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb3vOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbco2vv 2101 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑤]𝑤𝐴)
2 eleq1w 2899 . . . 4 (𝑤 = 𝑥 → (𝑤𝐴𝑥𝐴))
32sbievw 2096 . . 3 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
43sbbii 2074 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝐴 ↔ [𝑦 / 𝑥]𝑥𝐴)
5 eleq1w 2899 . . 3 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
65sbievw 2096 . 2 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
71, 4, 63bitr3i 302 1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207  [wsb 2062   ∈ wcel 2107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-sb 2063  df-clel 2897 This theorem is referenced by: (None)
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