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Theorem csbopg 4820
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 4525 . . 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 3825 . . . . 5 ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3 sbcel1g 4369 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ 𝐴 / 𝑥𝐶 ∈ V))
4 sbcel1g 4369 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ 𝐴 / 𝑥𝐷 ∈ V))
53, 4anbi12d 630 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
62, 5syl5bb 284 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V)))
7 csbprg 4644 . . . . 5 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}})
8 csbsng 4643 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶} = {𝐴 / 𝑥𝐶})
9 csbprg 4644 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝐶, 𝐷} = {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷})
108, 9preq12d 4676 . . . . 5 (𝐴𝑉 → {𝐴 / 𝑥{𝐶}, 𝐴 / 𝑥{𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
117, 10eqtrd 2861 . . . 4 (𝐴𝑉𝐴 / 𝑥{{𝐶}, {𝐶, 𝐷}} = {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}})
12 csbconstg 3906 . . . 4 (𝐴𝑉𝐴 / 𝑥∅ = ∅)
136, 11, 12ifbieq12d 4497 . . 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 2873 . 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 4799 . . 3 𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
1615csbeq2i 3895 . 2 𝐴 / 𝑥𝐶, 𝐷⟩ = 𝐴 / 𝑥if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17 dfopif 4799 . 2 𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩ = if((𝐴 / 𝑥𝐶 ∈ V ∧ 𝐴 / 𝑥𝐷 ∈ V), {{𝐴 / 𝑥𝐶}, {𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷}}, ∅)
1814, 16, 173eqtr4g 2886 1 (𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  [wsbc 3776  csb 3887  c0 4295  ifcif 4470  {csn 4564  {cpr 4566  cop 4570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571
This theorem is referenced by:  sbcop  5377  opsbc2ie  30172  esum2dlem  31256  csbfinxpg  34557
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