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Theorem eqsbc3rVD 39389
Description: Virtual deduction proof of eqsbc3r 3525. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eqsbc3rVD (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eqsbc3rVD
StepHypRef Expression
1 idn1 39107 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
2 eqsbc3 3508 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
31, 2e1a 39169 . . . . . 6 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶)   )
4 eqcom 2658 . . . . . . . . 9 (𝐶 = 𝑥𝑥 = 𝐶)
54sbcbiiOLD 39058 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
61, 5e1a 39169 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)   )
7 idn2 39155 . . . . . . 7 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
8 biimp 205 . . . . . . 7 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
96, 7, 8e12 39268 . . . . . 6 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
10 biimp 205 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
113, 9, 10e12 39268 . . . . 5 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐴 = 𝐶   )
12 eqcom 2658 . . . . 5 (𝐴 = 𝐶𝐶 = 𝐴)
1311, 12e2bi 39174 . . . 4 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐶 = 𝐴   )
1413in2 39147 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
15 idn2 39155 . . . . . . 7 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐶 = 𝐴   )
1615, 12e2bir 39175 . . . . . 6 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐴 = 𝐶   )
17 biimpr 210 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → (𝐴 = 𝐶[𝐴 / 𝑥]𝑥 = 𝐶))
183, 16, 17e12 39268 . . . . 5 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
19 biimpr 210 . . . . 5 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶[𝐴 / 𝑥]𝐶 = 𝑥))
206, 18, 19e12 39268 . . . 4 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
2120in2 39147 . . 3 (   𝐴𝐵   ▶   (𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥)   )
22 impbi 198 . . 3 (([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴) → ((𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥) → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)))
2314, 21, 22e11 39230 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
2423in1 39104 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 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  df-vd1 39103  df-vd2 39111
This theorem is referenced by: (None)
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