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Theorem bj-sbeqALT 34285
Description: Substitution in an equality (use the more general version bj-sbeq 34286 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 3890 . . 3 𝑥𝑦 / 𝑥𝐴
2 nfcsb1v 3890 . . 3 𝑥𝑦 / 𝑥𝐵
31, 2nfeq 2995 . 2 𝑥𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵
4 csbeq1a 3880 . . 3 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
5 csbeq1a 3880 . . 3 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
64, 5eqeq12d 2840 . 2 (𝑥 = 𝑦 → (𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵))
73, 6sbiev 2332 1 ([𝑦 / 𝑥]𝐴 = 𝐵𝑦 / 𝑥𝐴 = 𝑦 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  [wsb 2070  csb 3866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-sbc 3759  df-csb 3867
This theorem is referenced by: (None)
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