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Theorem bj-sbeqALT 33020
Description: Substitution in an equality (use the more genereal version bj-sbeq 33021 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 3582 . . 3 𝑥𝑦 / 𝑥𝐴
2 nfcsb1v 3582 . . 3 𝑥𝑦 / 𝑥𝐵
31, 2nfeq 2805 . 2 𝑥𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵
4 csbeq1a 3575 . . 3 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
5 csbeq1a 3575 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
64, 5eqeq12d 2666 . 2 (𝑥 = 𝑦 → (𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵))
73, 6sbie 2436 1 ([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  [wsb 1937  csb 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-sbc 3469  df-csb 3567
This theorem is referenced by: (None)
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