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Theorem csbsngVD 39645
Description: Virtual deduction proof of csbsng 4378. 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. csbsng 4378 is csbsngVD 39645 without virtual deductions and was automatically derived from csbsngVD 39645.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 3:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   ) 4:3: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 = ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 5:2,4: ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 6:5: ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵 ↔ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵)   ) 7:6: ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   ) 8:1: ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]𝑦 = 𝐵} = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   ) 9:7,8: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   ) 10:: ⊢ {𝐵} = {𝑦 ∣ 𝑦 = 𝐵} 11:10: ⊢ ∀𝑥{𝐵} = {𝑦 ∣ 𝑦 = 𝐵} 12:1,11: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = ⦋ 𝐴 / 𝑥⦌{𝑦 ∣ 𝑦 = 𝐵}   ) 13:9,12: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { 𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵}   ) 14:: ⊢ {⦋𝐴 / 𝑥⦌𝐵} = {𝑦 ∣ 𝑦 = ⦋𝐴 / 𝑥⦌𝐵} 15:13,14: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝐵} = { ⦋𝐴 / 𝑥⦌𝐵}   ) qed:15: ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐵} = {⦋ 𝐴 / 𝑥⦌𝐵})
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbsngVD (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})

Proof of Theorem csbsngVD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 idn1 39309 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbceqg 4126 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵))
31, 2e1a 39371 . . . . . . . 8 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵)   )
4 csbconstg 3693 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
51, 4e1a 39371 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
6 eqeq1 2774 . . . . . . . . 9 (𝐴 / 𝑥𝑦 = 𝑦 → (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵))
75, 6e1a 39371 . . . . . . . 8 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
8 bibi1 340 . . . . . . . . 9 (([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵) → (([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵) ↔ (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵)))
98biimprd 238 . . . . . . . 8 (([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵) → ((𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵) → ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)))
103, 7, 9e11 39432 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
1110gen11 39360 . . . . . 6 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
12 abbi 2885 . . . . . . 7 (∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵) ↔ {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
1312biimpi 206 . . . . . 6 (∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
1411, 13e1a 39371 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
15 csbabgOLD 39569 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦[𝐴 / 𝑥]𝑦 = 𝐵})
1615eqcomd 2776 . . . . . 6 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵})
171, 16e1a 39371 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
18 eqeq1 2774 . . . . . 6 ({𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵} → ({𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} ↔ 𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
1918biimpcd 239 . . . . 5 ({𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → ({𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵} → 𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
2014, 17, 19e11 39432 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
21 df-sn 4315 . . . . . 6 {𝐵} = {𝑦𝑦 = 𝐵}
2221ax-gen 1869 . . . . 5 𝑥{𝐵} = {𝑦𝑦 = 𝐵}
23 csbeq2gOLD 39284 . . . . 5 (𝐴𝑉 → (∀𝑥{𝐵} = {𝑦𝑦 = 𝐵} → 𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}))
241, 22, 23e10 39438 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
25 eqeq2 2781 . . . . 5 (𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵} ↔ 𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
2625biimpd 219 . . . 4 (𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵} → 𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
2720, 24, 26e11 39432 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
28 df-sn 4315 . . 3 {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
29 eqeq2 2781 . . . 4 ({𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵} ↔ 𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
3029biimprcd 240 . . 3 (𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → ({𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → 𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵}))
3127, 28, 30e10 39438 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵}   )
3231in1 39306 1 (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1628   = wceq 1630   ∈ wcel 2144  {cab 2756  [wsbc 3585  ⦋csb 3680  {csn 4314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-nul 4062  df-sn 4315  df-vd1 39305 This theorem is referenced by: (None)
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