Theorem csbopg 4358

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	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩)

Proof of Theorem csbopg

Step	Hyp	Ref	Expression
1		csbif 4088	. . 3 ⊢ ⦋𝐴 / 𝑥⦌if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅) = if([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V), ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}}, ⦋𝐴 / 𝑥⦌∅)
2		sbcan 3445	. . . . 5 ⊢ ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3		sbcel1g 3939	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ V))
4		sbcel1g 3939	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐷 ∈ V))
5	3, 4	anbi12d 743	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V) ↔ (⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V)))
6	2, 5	syl5bb 271	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V)))
7		csbprg 4191	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}})
8		csbsng 4190	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶} = {⦋𝐴 / 𝑥⦌𝐶})
9		csbprg 4191	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶, 𝐷} = {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷})
10	8, 9	preq12d 4220	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
11	7, 10	eqtrd 2644	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
12		csbconstg 3512	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∅ = ∅)
13	6, 11, 12	ifbieq12d 4063	. . 3 ⊢ (𝐴 ∈ 𝑉 → if([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V), ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}}, ⦋𝐴 / 𝑥⦌∅) = if((⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V), {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}}, ∅))
14	1, 13	syl5eq 2656	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅) = if((⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V), {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}}, ∅))
15		dfopif 4337	. . 3 ⊢ ⟨𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
16	15	csbeq2i 3945	. 2 ⊢ ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⦋𝐴 / 𝑥⦌if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17		dfopif 4337	. 2 ⊢ ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩ = if((⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V), {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}}, ∅)
18	14, 16, 17	3eqtr4g 2669	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874  ifcif 4036  {csn 4125  {cpr 4127  ⟨cop 4131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132

This theorem is referenced by:  esum2dlem  29481  csbfinxpg  32401