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Theorem csbopg 4891
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 4583 . . 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 3838 . . . . 5 ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3 sbcel1g 4416 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ 𝐴 / 𝑥𝐶 ∈ V))
4 sbcel1g 4416 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ 𝐴 / 𝑥𝐷 ∈ V))
53, 4anbi12d 632 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
62, 5bitrid 283 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
7 csbprg 4709 . . . . 5 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}})
8 csbsng 4708 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶} = {𝐴 / 𝑥𝐶})
9 csbprg 4709 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶, 𝐷} = {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷})
108, 9preq12d 4741 . . . . 5 (𝐴𝑉 → {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
117, 10eqtrd 2777 . . . 4 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
12 csbconstg 3918 . . . 4 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
136, 11, 12ifbieq12d 4554 . . 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, 13eqtrid 2789 . 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 4870 . . 3 𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
1615csbeq2i 3907 . 2 𝐴 / 𝑥𝐶, 𝐷⟩ = 𝐴 / 𝑥if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17 dfopif 4870 . 2 𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩ = if((𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V), {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}}, ∅)
1814, 16, 173eqtr4g 2802 1 (𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  csb 3899  c0 4333  ifcif 4525  {csn 4626  {cpr 4628  cop 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  sbcop  5494  opsbc2ie  32495  esum2dlem  34093  csbfinxpg  37389
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