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Theorem csbopg 4563
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 4274 . . 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 3611 . . . . 5 ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3 sbcel1g 4122 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ 𝐴 / 𝑥𝐶 ∈ V))
4 sbcel1g 4122 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ 𝐴 / 𝑥𝐷 ∈ V))
53, 4anbi12d 749 . . . . 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 4380 . . . . 5 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}})
8 csbsng 4379 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶} = {𝐴 / 𝑥𝐶})
9 csbprg 4380 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶, 𝐷} = {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷})
108, 9preq12d 4412 . . . . 5 (𝐴𝑉 → {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
117, 10eqtrd 2786 . . . 4 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
12 csbconstg 3679 . . . 4 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
136, 11, 12ifbieq12d 4249 . . 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 2798 . 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 4542 . . 3 𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
1615csbeq2i 4128 . 2 𝐴 / 𝑥𝐶, 𝐷⟩ = 𝐴 / 𝑥if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17 dfopif 4542 . 2 𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩ = if((𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V), {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}}, ∅)
1814, 16, 173eqtr4g 2811 1 (𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  Vcvv 3332  [wsbc 3568  csb 3666  c0 4050  ifcif 4222  {csn 4313  {cpr 4315  cop 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320
This theorem is referenced by:  esum2dlem  30455  csbfinxpg  33528
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