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Theorem eqsbc3r 2970
Description: eqsbc3 2949 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
Assertion
Ref Expression
eqsbc3r (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqsbc3 2949 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
2 eqcom 2142 . . 3 (𝐵 = 𝑥𝑥 = 𝐵)
32sbcbii 2969 . 2 ([𝐴 / 𝑥]𝐵 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐵)
4 eqcom 2142 . 2 (𝐵 = 𝐴𝐴 = 𝐵)
51, 3, 43bitr4g 222 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  wcel 1481  [wsbc 2910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-sbc 2911
This theorem is referenced by: (None)
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