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Theorem csbsngVD 43165
Description: Virtual deduction proof of csbsng 4669. 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 4669 is csbsngVD 43165 without virtual deductions and was automatically derived from csbsngVD 43165.
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 42846 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbceqg 4369 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵))
31, 2e1a 42899 . . . . . . . 8 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵)   )
4 csbconstg 3874 . . . . . . . . . 10 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
51, 4e1a 42899 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
6 eqeq1 2740 . . . . . . . . 9 (𝐴 / 𝑥𝑦 = 𝑦 → (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵))
75, 6e1a 42899 . . . . . . . 8 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
8 bibi1 351 . . . . . . . . 9 (([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵) → (([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵) ↔ (𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵)))
98biimprd 247 . . . . . . . 8 (([𝐴 / 𝑥]𝑦 = 𝐵𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵) → ((𝐴 / 𝑥𝑦 = 𝐴 / 𝑥𝐵𝑦 = 𝐴 / 𝑥𝐵) → ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)))
103, 7, 9e11 42960 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
1110gen11 42888 . . . . . 6 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)   )
12 abbi 2808 . . . . . . 7 (∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵) ↔ {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
1312biimpi 215 . . . . . 6 (∀𝑦([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
1411, 13e1a 42899 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
15 csbab 4397 . . . . . . . 8 𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦[𝐴 / 𝑥]𝑦 = 𝐵}
1615a1i 11 . . . . . . 7 (𝐴𝑉𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦[𝐴 / 𝑥]𝑦 = 𝐵})
1716eqcomd 2742 . . . . . 6 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵})
181, 17e1a 42899 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
19 eqeq1 2740 . . . . . 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}))
2019biimpcd 248 . . . . 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}))
2114, 18, 20e11 42960 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
22 df-sn 4587 . . . . . 6 {𝐵} = {𝑦𝑦 = 𝐵}
2322ax-gen 1797 . . . . 5 𝑥{𝐵} = {𝑦𝑦 = 𝐵}
24 csbeq2 3860 . . . . . 6 (∀𝑥{𝐵} = {𝑦𝑦 = 𝐵} → 𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵})
2524a1i 11 . . . . 5 (𝐴𝑉 → (∀𝑥{𝐵} = {𝑦𝑦 = 𝐵} → 𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}))
261, 23, 25e10 42966 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}   )
27 eqeq2 2748 . . . . 5 (𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵} ↔ 𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
2827biimpd 228 . . . 4 (𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵} → 𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
2921, 26, 28e11 42960 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}   )
30 df-sn 4587 . . 3 {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
31 eqeq2 2748 . . . 4 ({𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵} ↔ 𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}))
3231biimprcd 249 . . 3 (𝐴 / 𝑥{𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → ({𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵} → 𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵}))
3329, 30, 32e10 42966 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵}   )
3433in1 42843 1 (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wcel 2106  {cab 2713  [wsbc 3739  csb 3855  {csn 4586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-nul 4283  df-sn 4587  df-vd1 42842
This theorem is referenced by: (None)
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