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Theorem eqsbc3rVD 41041
Description: Virtual deduction proof of eqsbc3r 3840. (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 40775 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
2 eqsbc3 3820 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
31, 2e1a 40828 . . . . . 6 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶)   )
4 eqcom 2832 . . . . . . . . . 10 (𝐶 = 𝑥𝑥 = 𝐶)
54sbcbii 3832 . . . . . . . . 9 ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)
65a1i 11 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
71, 6e1a 40828 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)   )
8 idn2 40814 . . . . . . 7 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
9 biimp 216 . . . . . . 7 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
107, 8, 9e12 40925 . . . . . 6 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
11 biimp 216 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
123, 10, 11e12 40925 . . . . 5 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐴 = 𝐶   )
13 eqcom 2832 . . . . 5 (𝐴 = 𝐶𝐶 = 𝐴)
1412, 13e2bi 40833 . . . 4 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐶 = 𝐴   )
1514in2 40806 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
16 idn2 40814 . . . . . . 7 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐶 = 𝐴   )
1716, 13e2bir 40834 . . . . . 6 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐴 = 𝐶   )
18 biimpr 221 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → (𝐴 = 𝐶[𝐴 / 𝑥]𝑥 = 𝐶))
193, 17, 18e12 40925 . . . . 5 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
20 biimpr 221 . . . . 5 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶[𝐴 / 𝑥]𝐶 = 𝑥))
217, 19, 20e12 40925 . . . 4 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
2221in2 40806 . . 3 (   𝐴𝐵   ▶   (𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥)   )
23 impbi 209 . . 3 (([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴) → ((𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥) → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)))
2415, 22, 23e11 40889 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
2524in1 40772 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2107  [wsbc 3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-sbc 3776  df-vd1 40771  df-vd2 40779
This theorem is referenced by: (None)
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