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Theorem bj-sbeqALT 36883
Description: Substitution in an equality (use the more general version bj-sbeq 36884 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 3933 . . 3 𝑥𝑦 / 𝑥𝐴
2 nfcsb1v 3933 . . 3 𝑥𝑦 / 𝑥𝐵
31, 2nfeq 2917 . 2 𝑥𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵
4 csbeq1a 3922 . . 3 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
5 csbeq1a 3922 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
64, 5eqeq12d 2751 . 2 (𝑥 = 𝑦 → (𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵))
73, 6sbiev 2313 1 ([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  [wsb 2062  csb 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-sbc 3792  df-csb 3909
This theorem is referenced by: (None)
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