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Theorem eqsbc3rOLD 3526
 Description: Obsolete proof of eqsbc3r 3525 as of 7-Jul-2021. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 39389 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
eqsbc3rOLD (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc3rOLD
StepHypRef Expression
1 eqcom 2658 . . . . . 6 (𝐵 = 𝑥𝑥 = 𝐵)
21sbcbii 3524 . . . . 5 ([𝐴 / 𝑥]𝐵 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐵)
32biimpi 206 . . . 4 ([𝐴 / 𝑥]𝐵 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐵)
4 eqsbc3 3508 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
53, 4syl5ib 234 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐴 = 𝐵))
6 eqcom 2658 . . 3 (𝐴 = 𝐵𝐵 = 𝐴)
75, 6syl6ib 241 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
8 idd 24 . . . . 5 (𝐴𝑉 → (𝐵 = 𝐴𝐵 = 𝐴))
98, 6syl6ibr 242 . . . 4 (𝐴𝑉 → (𝐵 = 𝐴𝐴 = 𝐵))
109, 4sylibrd 249 . . 3 (𝐴𝑉 → (𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵))
1110, 2syl6ibr 242 . 2 (𝐴𝑉 → (𝐵 = 𝐴[𝐴 / 𝑥]𝐵 = 𝑥))
127, 11impbid 202 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523   ∈ wcel 2030  [wsbc 3468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469 This theorem is referenced by: (None)
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