Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcel12 Structured version   Visualization version   GIF version

Theorem sbcel12 4308
 Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcel12 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Proof of Theorem sbcel12
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3701 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵𝐶[𝐴 / 𝑥]𝐵𝐶))
2 dfsbcq2 3701 . . . . . 6 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐵))
32abbidv 2822 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐵})
4 dfsbcq2 3701 . . . . . 6 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2822 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
63, 5eleq12d 2846 . . . 4 (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
7 nfs1v 2157 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑦𝐵
87nfab 2925 . . . . . 6 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵}
9 nfs1v 2157 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑦𝐶
109nfab 2925 . . . . . 6 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
118, 10nfel 2933 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
12 sbab 2898 . . . . . 6 (𝑥 = 𝑧𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵})
13 sbab 2898 . . . . . 6 (𝑥 = 𝑧𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
1412, 13eleq12d 2846 . . . . 5 (𝑥 = 𝑧 → (𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}))
1511, 14sbiev 2322 . . . 4 ([𝑧 / 𝑥]𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
161, 6, 15vtoclbg 3489 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
17 df-csb 3808 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
18 df-csb 3808 . . . 4 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
1917, 18eleq12i 2844 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶})
2016, 19bitr4di 292 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
21 sbcex 3708 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
2221con3i 157 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
23 noel 4232 . . . 4 ¬ 𝐴 / 𝑥𝐵 ∈ ∅
24 csbprc 4305 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
2524eleq2d 2837 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵 ∈ ∅))
2623, 25mtbiri 330 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2722, 262falsed 380 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2820, 27pm2.61i 185 1 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538  [wsb 2069   ∈ wcel 2111  {cab 2735  Vcvv 3409  [wsbc 3698  ⦋csb 3807  ∅c0 4227 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-nul 4228 This theorem is referenced by:  sbcnel12g  4311  sbcel1g  4313  sbcel2  4315  sbccsb2  4334  csbmpt12  5418  ixpsnval  8495  fmptdF  30530  csbmpo123  35063  csbfinxpg  35120  finixpnum  35357  csbxpgVD  42018  csbrngVD  42020
 Copyright terms: Public domain W3C validator