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Theorem csbeq2gVD 39627
Description: Virtual deduction proof of csbeq2gOLD 39267. 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. csbeq2gOLD 39267 is csbeq2gVD 39627 without virtual deductions and was automatically derived from csbeq2gVD 39627.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → [𝐴 / 𝑥] 𝐵 = 𝐶)   ) 3:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   ) 4:2,3: ⊢ (   𝐴 ∈ 𝑉   ▶   (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥 ⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)   ) qed:4: ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝐵 = 𝐶 → ⦋𝐴 / 𝑥⦌ 𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
(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 39292 . . . 4 (   𝐴𝑉   ▶   𝐴𝑉   )
2 spsbc 3589 . . . 4 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
31, 2e1a 39354 . . 3 (   𝐴𝑉   ▶   (∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶)   )
4 sbceqg 4127 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
51, 4e1a 39354 . . 3 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)   )
6 imbi2 337 . . . 4 (([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶) → ((∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶) ↔ (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
76biimpcd 239 . . 3 ((∀𝑥 𝐵 = 𝐶[𝐴 / 𝑥]𝐵 = 𝐶) → (([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶) → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
83, 5, 7e11 39415 . 2 (   𝐴𝑉   ▶   (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)   )
98in1 39289 1 (𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1630   = wceq 1632   ∈ wcel 2139  [wsbc 3576  ⦋csb 3674 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675  df-vd1 39288 This theorem is referenced by: (None)
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