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Theorem csbeq2gVD 44229
Description: Virtual deduction proof of csbeq2 3893. 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 3893 is csbeq2gVD 44229 without virtual deductions and was automatically derived from csbeq2gVD 44229.
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 43911 . . . 4 (   𝐴𝑉   ▶   𝐴𝑉   )
2 spsbc 3785 . . . 4 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
31, 2e1a 43964 . . 3 (   𝐴𝑉   ▶   (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶)   )
4 sbceqg 4404 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
51, 4e1a 43964 . . 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 248 . . 3 ((∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶) → (([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶) → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
83, 5, 7e11 44025 . 2 (   𝐴𝑉   ▶   (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)   )
98in1 43908 1 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wcel 2098  [wsbc 3772  csb 3888
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-sbc 3773  df-csb 3889  df-vd1 43907
This theorem is referenced by: (None)
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