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Theorem eqsbc2VD 43903
Description: Virtual deduction proof of eqsbc2 3845. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
eqsbc2VD (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eqsbc2VD
StepHypRef Expression
1 idn1 43637 . . . . . . 7 (   𝐴𝐵   ▶   𝐴𝐵   )
2 eqsbc1 3825 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
31, 2e1a 43690 . . . . . 6 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶)   )
4 eqcom 2737 . . . . . . . . . 10 (𝐶 = 𝑥𝑥 = 𝐶)
54sbcbii 3836 . . . . . . . . 9 ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)
65a1i 11 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
71, 6e1a 43690 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶)   )
8 idn2 43676 . . . . . . 7 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
9 biimp 214 . . . . . . 7 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶))
107, 8, 9e12 43787 . . . . . 6 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
11 biimp 214 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶))
123, 10, 11e12 43787 . . . . 5 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐴 = 𝐶   )
13 eqcom 2737 . . . . 5 (𝐴 = 𝐶𝐶 = 𝐴)
1412, 13e2bi 43695 . . . 4 (   𝐴𝐵   ,   [𝐴 / 𝑥]𝐶 = 𝑥   ▶   𝐶 = 𝐴   )
1514in2 43668 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
16 idn2 43676 . . . . . . 7 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐶 = 𝐴   )
1716, 13e2bir 43696 . . . . . 6 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   𝐴 = 𝐶   )
18 biimpr 219 . . . . . 6 (([𝐴 / 𝑥]𝑥 = 𝐶𝐴 = 𝐶) → (𝐴 = 𝐶[𝐴 / 𝑥]𝑥 = 𝐶))
193, 17, 18e12 43787 . . . . 5 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝑥 = 𝐶   )
20 biimpr 219 . . . . 5 (([𝐴 / 𝑥]𝐶 = 𝑥[𝐴 / 𝑥]𝑥 = 𝐶) → ([𝐴 / 𝑥]𝑥 = 𝐶[𝐴 / 𝑥]𝐶 = 𝑥))
217, 19, 20e12 43787 . . . 4 (   𝐴𝐵   ,   𝐶 = 𝐴   ▶   [𝐴 / 𝑥]𝐶 = 𝑥   )
2221in2 43668 . . 3 (   𝐴𝐵   ▶   (𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥)   )
23 impbi 207 . . 3 (([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴) → ((𝐶 = 𝐴[𝐴 / 𝑥]𝐶 = 𝑥) → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)))
2415, 22, 23e11 43751 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴)   )
2524in1 43634 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  [wsbc 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-sbc 3777  df-vd1 43633  df-vd2 43641
This theorem is referenced by: (None)
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