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

Theorem sbcel12 4408
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 3780 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵𝐶[𝐴 / 𝑥]𝐵𝐶))
2 dfsbcq2 3780 . . . . . 6 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐵))
32abbidv 2800 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐵})
4 dfsbcq2 3780 . . . . . 6 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2800 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
63, 5eleq12d 2826 . . . 4 (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
7 nfs1v 2152 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑦𝐵
87nfab 2908 . . . . . 6 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵}
9 nfs1v 2152 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑦𝐶
109nfab 2908 . . . . . 6 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
118, 10nfel 2916 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
12 sbab 2881 . . . . . 6 (𝑥 = 𝑧𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵})
13 sbab 2881 . . . . . 6 (𝑥 = 𝑧𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
1412, 13eleq12d 2826 . . . . 5 (𝑥 = 𝑧 → (𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}))
1511, 14sbiev 2307 . . . 4 ([𝑧 / 𝑥]𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
161, 6, 15vtoclbg 3544 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
17 df-csb 3894 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
18 df-csb 3894 . . . 4 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
1917, 18eleq12i 2825 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶})
2016, 19bitr4di 289 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
21 sbcex 3787 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
2221con3i 154 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
23 noel 4330 . . . 4 ¬ 𝐴 / 𝑥𝐵 ∈ ∅
24 csbprc 4406 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
2524eleq2d 2818 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵 ∈ ∅))
2623, 25mtbiri 327 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2722, 262falsed 376 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2820, 27pm2.61i 182 1 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1540  [wsb 2066  wcel 2105  {cab 2708  Vcvv 3473  [wsbc 3777  csb 3893  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-nul 4323
This theorem is referenced by:  sbcnel12g  4411  sbcel1g  4413  sbcel2  4415  sbccsb2  4434  csbmpt12  5557  ixpsnval  8897  fmptdF  32149  csbmpo123  36516  csbfinxpg  36573  finixpnum  36777  csbxpgVD  43958  csbrngVD  43960
  Copyright terms: Public domain W3C validator