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

Theorem csbopg 4611
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 4332 . . 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 3674 . . . . 5 ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3 sbcel1g 4182 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ 𝐴 / 𝑥𝐶 ∈ V))
4 sbcel1g 4182 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ 𝐴 / 𝑥𝐷 ∈ V))
53, 4anbi12d 618 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
62, 5syl5bb 274 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
7 csbprg 4434 . . . . 5 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}})
8 csbsng 4433 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶} = {𝐴 / 𝑥𝐶})
9 csbprg 4434 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶, 𝐷} = {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷})
108, 9preq12d 4465 . . . . 5 (𝐴𝑉 → {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
117, 10eqtrd 2838 . . . 4 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
12 csbconstg 3739 . . . 4 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
136, 11, 12ifbieq12d 4304 . . 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 2850 . 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 4590 . . 3 𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
1615csbeq2i 4188 . 2 𝐴 / 𝑥𝐶, 𝐷⟩ = 𝐴 / 𝑥if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17 dfopif 4590 . 2 𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩ = if((𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V), {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}}, ∅)
1814, 16, 173eqtr4g 2863 1 (𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  Vcvv 3389  [wsbc 3631  csb 3726  c0 4114  ifcif 4277  {csn 4368  {cpr 4370  cop 4374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375
This theorem is referenced by:  esum2dlem  30477  csbfinxpg  33539
  Copyright terms: Public domain W3C validator