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

Theorem csbuni 4935
Description: Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
csbuni 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵

Proof of Theorem csbuni
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbab 4439 . . . 4 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}
2 sbcex2 3849 . . . . . 6 ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))
3 sbcan 3837 . . . . . . . 8 ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))
4 sbcg 3862 . . . . . . . . . 10 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
54anbi1d 631 . . . . . . . . 9 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
6 sbcel2 4417 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)
76anbi2i 623 . . . . . . . . 9 ((𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))
85, 7bitrdi 287 . . . . . . . 8 (𝐴 ∈ V → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
93, 8bitrid 283 . . . . . . 7 (𝐴 ∈ V → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
109exbidv 1920 . . . . . 6 (𝐴 ∈ V → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
112, 10bitrid 283 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
1211abbidv 2807 . . . 4 (𝐴 ∈ V → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
131, 12eqtrid 2788 . . 3 (𝐴 ∈ V → 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
14 df-uni 4907 . . . 4 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
1514csbeq2i 3906 . . 3 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
16 df-uni 4907 . . 3 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
1713, 15, 163eqtr4g 2801 . 2 (𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
18 csbprc 4408 . . 3 𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = ∅)
19 csbprc 4408 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
2019unieqd 4919 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
21 uni0 4934 . . . 4 ∅ = ∅
2220, 21eqtr2di 2793 . . 3 𝐴 ∈ V → ∅ = 𝐴 / 𝑥𝐵)
2318, 22eqtrd 2776 . 2 𝐴 ∈ V → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
2417, 23pm2.61i 182 1 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1539  wex 1778  wcel 2107  {cab 2713  Vcvv 3479  [wsbc 3787  csb 3898  c0 4332   cuni 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-ss 3967  df-nul 4333  df-sn 4626  df-uni 4907
This theorem is referenced by:  csbfrecsg  8310  csbfv12gALTVD  44924
  Copyright terms: Public domain W3C validator