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

Theorem csbopg 4451
 Description: Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.) (Revised by ML, 25-Oct-2020.)
Assertion
Ref Expression
csbopg (𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)

Proof of Theorem csbopg
StepHypRef Expression
1 csbif 4171 . . 3 𝐴 / 𝑥if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅) = if([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V), 𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}}, 𝐴 / 𝑥∅)
2 sbcan 3511 . . . . 5 ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3 sbcel1g 4020 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ 𝐴 / 𝑥𝐶 ∈ V))
4 sbcel1g 4020 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ 𝐴 / 𝑥𝐷 ∈ V))
53, 4anbi12d 747 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
62, 5syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
7 csbprg 4276 . . . . 5 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}})
8 csbsng 4275 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶} = {𝐴 / 𝑥𝐶})
9 csbprg 4276 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶, 𝐷} = {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷})
108, 9preq12d 4308 . . . . 5 (𝐴𝑉 → {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
117, 10eqtrd 2685 . . . 4 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
12 csbconstg 3579 . . . 4 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
136, 11, 12ifbieq12d 4146 . . 3 (𝐴𝑉 → if([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V), 𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}}, 𝐴 / 𝑥∅) = if((𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V), {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}}, ∅))
141, 13syl5eq 2697 . 2 (𝐴𝑉𝐴 / 𝑥if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅) = if((𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V), {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}}, ∅))
15 dfopif 4430 . . 3 𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
1615csbeq2i 4026 . 2 𝐴 / 𝑥𝐶, 𝐷⟩ = 𝐴 / 𝑥if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17 dfopif 4430 . 2 𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩ = if((𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V), {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}}, ∅)
1814, 16, 173eqtr4g 2710 1 (𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  [wsbc 3468  ⦋csb 3566  ∅c0 3948  ifcif 4119  {csn 4210  {cpr 4212  ⟨cop 4216 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217 This theorem is referenced by:  esum2dlem  30282  csbfinxpg  33355
 Copyright terms: Public domain W3C validator