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Theorem clelsb2OLD 2861
Description: Obsolete version of clelsb2 2860 as of 24-Nov-2024.) (Contributed by Jim Kingdon, 22-Nov-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
clelsb2OLD ([𝑦 / 𝑥]𝐴𝑥𝐴𝑦)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb2OLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1916 . . 3 𝑥 𝐴𝑤
21sbco2 2509 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴𝑤 ↔ [𝑦 / 𝑤]𝐴𝑤)
3 nfv 1916 . . . 4 𝑤 𝐴𝑥
4 eleq2 2821 . . . 4 (𝑤 = 𝑥 → (𝐴𝑤𝐴𝑥))
53, 4sbie 2500 . . 3 ([𝑥 / 𝑤]𝐴𝑤𝐴𝑥)
65sbbii 2078 . 2 ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴𝑤 ↔ [𝑦 / 𝑥]𝐴𝑥)
7 nfv 1916 . . 3 𝑤 𝐴𝑦
8 eleq2 2821 . . 3 (𝑤 = 𝑦 → (𝐴𝑤𝐴𝑦))
97, 8sbie 2500 . 2 ([𝑦 / 𝑤]𝐴𝑤𝐴𝑦)
102, 6, 93bitr3i 301 1 ([𝑦 / 𝑥]𝐴𝑥𝐴𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2066  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-cleq 2723  df-clel 2809
This theorem is referenced by: (None)
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