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

Proof of Theorem csbopg

Step	Hyp	Ref	Expression
1		csbif 4583	. . 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 3838	. . . . 5 ⊢ ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3		sbcel1g 4416	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ V))
4		sbcel1g 4416	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐷 ∈ V))
5	3, 4	anbi12d 632	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V) ↔ (⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V)))
6	2, 5	bitrid 283	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V)))
7		csbprg 4709	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}})
8		csbsng 4708	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶} = {⦋𝐴 / 𝑥⦌𝐶})
9		csbprg 4709	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶, 𝐷} = {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷})
10	8, 9	preq12d 4741	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
11	7, 10	eqtrd 2777	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
12		csbconstg 3918	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∅ = ∅)
13	6, 11, 12	ifbieq12d 4554	. . 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	eqtrid 2789	. 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 4870	. . 3 ⊢ ⟨𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
16	15	csbeq2i 3907	. 2 ⊢ ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⦋𝐴 / 𝑥⦌if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17		dfopif 4870	. 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 2802	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩)

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