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Theorem eqsbc3rVD 41589
 Description: Virtual deduction proof of eqsbc3r 3784. (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 41323 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
2 eqsbc3 3765 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
31, 2e1a 41376 . . . . . 6 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶)   )
4 eqcom 2805 . . . . . . . . . 10 (𝐶 = 𝑥𝑥 = 𝐶)
54sbcbii 3776 . . . . . . . . 9 ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)
65a1i 11 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
71, 6e1a 41376 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)   )
8 idn2 41362 . . . . . . 7 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
9 biimp 218 . . . . . . 7 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
107, 8, 9e12 41473 . . . . . 6 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
11 biimp 218 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
123, 10, 11e12 41473 . . . . 5 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐴 = 𝐶   )
13 eqcom 2805 . . . . 5 (𝐴 = 𝐶𝐶 = 𝐴)
1412, 13e2bi 41381 . . . 4 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐶 = 𝐴   )
1514in2 41354 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
16 idn2 41362 . . . . . . 7 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐶 = 𝐴   )
1716, 13e2bir 41382 . . . . . 6 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐴 = 𝐶   )
18 biimpr 223 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → (𝐴 = 𝐶[𝐴 / 𝑥]𝑥 = 𝐶))
193, 17, 18e12 41473 . . . . 5 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
20 biimpr 223 . . . . 5 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶[𝐴 / 𝑥]𝐶 = 𝑥))
217, 19, 20e12 41473 . . . 4 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
2221in2 41354 . . 3 (   𝐴𝐵   ▶   (𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥)   )
23 impbi 211 . . 3 (([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴) → ((𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥) → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)))
2415, 22, 23e11 41437 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
2524in1 41320 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  [wsbc 3720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-sbc 3721  df-vd1 41319  df-vd2 41327 This theorem is referenced by: (None)
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