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Theorem sbcel12 4357
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 3772 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵𝐶[𝐴 / 𝑥]𝐵𝐶))
2 dfsbcq2 3772 . . . . . 6 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐵))
32abbidv 2882 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐵})
4 dfsbcq2 3772 . . . . . 6 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2882 . . . . 5 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
63, 5eleq12d 2904 . . . 4 (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
7 nfs1v 2264 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑦𝐵
87nfab 2981 . . . . . 6 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵}
9 nfs1v 2264 . . . . . . 7 𝑥[𝑧 / 𝑥]𝑦𝐶
109nfab 2981 . . . . . 6 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
118, 10nfel 2989 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
12 sbab 2957 . . . . . 6 (𝑥 = 𝑧𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵})
13 sbab 2957 . . . . . 6 (𝑥 = 𝑧𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
1412, 13eleq12d 2904 . . . . 5 (𝑥 = 𝑧 → (𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}))
1511, 14sbiev 2321 . . . 4 ([𝑧 / 𝑥]𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
161, 6, 15vtoclbg 3566 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
17 df-csb 3881 . . . 4 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
18 df-csb 3881 . . . 4 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
1917, 18eleq12i 2902 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶})
2016, 19syl6bbr 290 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
21 sbcex 3779 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
2221con3i 157 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
23 noel 4293 . . . 4 ¬ 𝐴 / 𝑥𝐵 ∈ ∅
24 csbprc 4355 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
2524eleq2d 2895 . . . 4 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵 ∈ ∅))
2623, 25mtbiri 328 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
2722, 262falsed 378 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
2820, 27pm2.61i 183 1 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  [wsb 2060  wcel 2105  {cab 2796  Vcvv 3492  [wsbc 3769  csb 3880  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-nul 4289
This theorem is referenced by:  sbcnel12g  4360  sbcel1g  4362  sbcel2  4364  sbccsb2  4383  csbmpt12  5435  ixpsnval  8452  fmptdF  30329  csbmpo123  34494  csbfinxpg  34551  finixpnum  34758  csbxpgVD  41105  csbrngVD  41107
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