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Theorem csbsngVD 39628
Description: Virtual deduction proof of csbsng 4387. 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 4387 is csbsngVD 39628 without virtual deductions and was automatically derived from csbsngVD 39628.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵)   )
3:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
4:3: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
5:2,4: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵 𝑦 = 𝐴 / 𝑥𝐵)   )
6:5: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
7:6: (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
8:1: (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
9:7,8: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
10:: {𝐵} = {𝑦𝑦 = 𝐵}
11:10: 𝑥{𝐵} = {𝑦𝑦 = 𝐵}
12:1,11: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
13:9,12: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = { 𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
14:: {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
15:13,14: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = { 𝐴 / 𝑥𝐵}   )
qed:15: (𝐴𝑉𝐴 / 𝑥{𝐵} = { 𝐴 / 𝑥𝐵})
(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 39292 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbceqg 4127 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵))
31, 2e1a 39354 . . . . . . . 8 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵)   )
4 csbconstg 3687 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
51, 4e1a 39354 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
6 eqeq1 2764 . . . . . . . . 9 (𝐴 / 𝑥𝑦 = 𝑦 → (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵))
75, 6e1a 39354 . . . . . . . 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 39415 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
1110gen11 39343 . . . . . 6 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
12 abbi 2875 . . . . . . 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 39354 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
15 csbabgOLD 39552 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦[𝐴 / 𝑥]𝑦 = 𝐵})
1615eqcomd 2766 . . . . . 6 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵})
171, 16e1a 39354 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
18 eqeq1 2764 . . . . . 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 39415 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
21 df-sn 4322 . . . . . 6 {𝐵} = {𝑦𝑦 = 𝐵}
2221ax-gen 1871 . . . . 5 𝑥{𝐵} = {𝑦𝑦 = 𝐵}
23 csbeq2gOLD 39267 . . . . 5 (𝐴𝑉 → (∀𝑥{𝐵} = {𝑦𝑦 = 𝐵} → 𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}))
241, 22, 23e10 39421 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
25 eqeq2 2771 . . . . 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 39415 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
28 df-sn 4322 . . 3 {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
29 eqeq2 2771 . . . 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 39421 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵}   )
3231in1 39289 1 (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630   = wceq 1632  wcel 2139  {cab 2746  [wsbc 3576  csb 3674  {csn 4321
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-fal 1638  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-dif 3718  df-nul 4059  df-sn 4322  df-vd1 39288
This theorem is referenced by: (None)
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