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Theorem eqsbc3r 2845
Description: eqsbc3 2824 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 2824 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
2 eqcom 2058 . . 3 (𝐵 = 𝑥𝑥 = 𝐵)
32sbcbii 2844 . 2 ([𝐴 / 𝑥]𝐵 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐵)
4 eqcom 2058 . 2 (𝐵 = 𝐴𝐴 = 𝐵)
51, 3, 43bitr4g 216 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  [wsbc 2786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2787
This theorem is referenced by: (None)
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