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Theorem wl-clelsb3df 34730
Description: Deduction version of clelsb3f 2986. (Contributed by Wolf Lammen, 29-May-2023.)
Hypotheses
Ref Expression
clelsb3df.1 𝑦𝜑
clelsb3df.2 (𝜑𝑦𝐴)
Assertion
Ref Expression
wl-clelsb3df (𝜑 → ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴))

Proof of Theorem wl-clelsb3df
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1908 . . 3 𝑤𝜑
2 clelsb3df.1 . . 3 𝑦𝜑
3 clelsb3df.2 . . . 4 (𝜑𝑦𝐴)
43nfcrd 2973 . . 3 (𝜑 → Ⅎ𝑦 𝑤𝐴)
51, 2, 4sbco2d 2552 . 2 (𝜑 → ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑤]𝑤𝐴))
6 clelsb3 2944 . . 3 ([𝑦 / 𝑤]𝑤𝐴𝑦𝐴)
76sbbii 2074 . 2 ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
8 clelsb3 2944 . 2 ([𝑥 / 𝑤]𝑤𝐴𝑥𝐴)
95, 7, 83bitr3g 314 1 (𝜑 → ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wnf 1777  [wsb 2062  wcel 2107  wnfc 2965
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  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clel 2897  df-nfc 2967
This theorem is referenced by:  wl-dfrabf  34731
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