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Theorem eqsbc3r 2819
 Description: eqsbc3 2802 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
eqsbc3r (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqcom 2042 . . . . . 6 (𝐶 = 𝑥𝑥 = 𝐶)
21sbcbii 2818 . . . . 5 ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)
32biimpi 113 . . . 4 ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)
4 eqsbc3 2802 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
53, 4syl5ib 143 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐴 = 𝐶))
6 eqcom 2042 . . 3 (𝐴 = 𝐶𝐶 = 𝐴)
75, 6syl6ib 150 . 2 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
8 idd 21 . . . . 5 (𝐴𝐵 → (𝐶 = 𝐴𝐶 = 𝐴))
98, 6syl6ibr 151 . . . 4 (𝐴𝐵 → (𝐶 = 𝐴𝐴 = 𝐶))
109, 4sylibrd 158 . . 3 (𝐴𝐵 → (𝐶 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐶))
1110, 2syl6ibr 151 . 2 (𝐴𝐵 → (𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥))
127, 11impbid 120 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  [wsbc 2764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765 This theorem is referenced by: (None)
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