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Theorem bj-sbeqALT 31919
Description: Substitution in an equality (use the more genereal version bj-sbeq 31920 instead, without disjoint variable condition). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sbeqALT ([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem bj-sbeqALT
StepHypRef Expression
1 nfcsb1v 3419 . . 3 𝑥𝑦 / 𝑥𝐴
2 nfcsb1v 3419 . . 3 𝑥𝑦 / 𝑥𝐵
31, 2nfeq 2666 . 2 𝑥𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵
4 csbeq1a 3412 . . 3 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
5 csbeq1a 3412 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
64, 5eqeq12d 2529 . 2 (𝑥 = 𝑦 → (𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵))
73, 6sbie 2300 1 ([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  [wsb 1830  csb 3403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-sbc 3307  df-csb 3404
This theorem is referenced by: (None)
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