Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcel12gOLD Structured version   Visualization version   GIF version

Theorem sbcel12gOLD 38574
Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcel12 3974 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcel12gOLD (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem sbcel12gOLD
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3432 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝐵𝐶[𝐴 / 𝑥]𝐵𝐶))
2 dfsbcq2 3432 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐵[𝐴 / 𝑥]𝑦𝐵))
32abbidv 2739 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} = {𝑦[𝐴 / 𝑥]𝑦𝐵})
4 dfsbcq2 3432 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐶))
54abbidv 2739 . . . 4 (𝑧 = 𝐴 → {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} = {𝑦[𝐴 / 𝑥]𝑦𝐶})
63, 5eleq12d 2693 . . 3 (𝑧 = 𝐴 → ({𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶} ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
7 nfs1v 2435 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐵
87nfab 2766 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵}
9 nfs1v 2435 . . . . . 6 𝑥[𝑧 / 𝑥]𝑦𝐶
109nfab 2766 . . . . 5 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
118, 10nfel 2774 . . . 4 𝑥{𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}
12 sbab 2748 . . . . 5 (𝑥 = 𝑧𝐵 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵})
13 sbab 2748 . . . . 5 (𝑥 = 𝑧𝐶 = {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
1412, 13eleq12d 2693 . . . 4 (𝑥 = 𝑧 → (𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶}))
1511, 14sbie 2406 . . 3 ([𝑧 / 𝑥]𝐵𝐶 ↔ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐵} ∈ {𝑦 ∣ [𝑧 / 𝑥]𝑦𝐶})
161, 6, 15vtoclbg 3262 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶}))
17 df-csb 3527 . . 3 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
18 df-csb 3527 . . 3 𝐴 / 𝑥𝐶 = {𝑦[𝐴 / 𝑥]𝑦𝐶}
1917, 18eleq12i 2692 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ {𝑦[𝐴 / 𝑥]𝑦𝐵} ∈ {𝑦[𝐴 / 𝑥]𝑦𝐶})
2016, 19syl6bbr 278 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  [wsb 1878  wcel 1988  {cab 2606  [wsbc 3429  csb 3526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-sbc 3430  df-csb 3527
This theorem is referenced by:  sbcel2gOLD  38575  csbxpgVD  38950  csbrngVD  38952
  Copyright terms: Public domain W3C validator