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Theorem bj-clelsb3 32548
Description: Remove dependency on ax-ext 2601 (and df-cleq 2614) from clelsb3 2726. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-clelsb3 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bj-clelsb3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . 3 𝑦 𝑧𝐴
21sbco2 2414 . 2 ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧𝐴 ↔ [𝑥 / 𝑧]𝑧𝐴)
3 nfv 1840 . . . 4 𝑧 𝑦𝐴
4 bj-eleq1w 32546 . . . 4 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
53, 4sbie 2407 . . 3 ([𝑦 / 𝑧]𝑧𝐴𝑦𝐴)
65sbbii 1884 . 2 ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧𝐴 ↔ [𝑥 / 𝑦]𝑦𝐴)
7 nfv 1840 . . 3 𝑧 𝑥𝐴
8 bj-eleq1w 32546 . . 3 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
97, 8sbie 2407 . 2 ([𝑥 / 𝑧]𝑧𝐴𝑥𝐴)
102, 6, 93bitr3i 290 1 ([𝑥 / 𝑦]𝑦𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1877  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clel 2617
This theorem is referenced by:  bj-hblem  32549
  Copyright terms: Public domain W3C validator