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Theorem csbeq2gVD 44890
Description: Virtual deduction proof of csbeq2 3913. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbeq2 3913 is csbeq2gVD 44890 without virtual deductions and was automatically derived from csbeq2gVD 44890.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   (∀𝑥𝐵 = 𝐶[𝐴 / 𝑥] 𝐵 = 𝐶)   )
3:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)   )
4:2,3: (   𝐴𝑉   ▶   (∀𝑥𝐵 = 𝐶𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐶)   )
qed:4: (𝐴𝑉 → (∀𝑥𝐵 = 𝐶𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbeq2gVD (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Proof of Theorem csbeq2gVD
StepHypRef Expression
1 idn1 44572 . . . 4 (   𝐴𝑉   ▶   𝐴𝑉   )
2 spsbc 3804 . . . 4 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
31, 2e1a 44625 . . 3 (   𝐴𝑉   ▶   (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶)   )
4 sbceqg 4418 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
51, 4e1a 44625 . . 3 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)   )
6 imbi2 348 . . . 4 (([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶) → ((∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶) ↔ (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
76biimpcd 249 . . 3 ((∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶) → (([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶) → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
83, 5, 7e11 44686 . 2 (   𝐴𝑉   ▶   (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)   )
98in1 44569 1 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wcel 2106  [wsbc 3791  csb 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-sbc 3792  df-csb 3909  df-vd1 44568
This theorem is referenced by: (None)
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