Theorem csbopg 4867

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 4558	. . 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 3815	. . . . 5 ⊢ ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3		sbcel1g 4391	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ V))
4		sbcel1g 4391	. . . . . 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 4685	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}})
8		csbsng 4684	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶} = {⦋𝐴 / 𝑥⦌𝐶})
9		csbprg 4685	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶, 𝐷} = {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷})
10	8, 9	preq12d 4717	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
11	7, 10	eqtrd 2770	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
12		csbconstg 3893	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∅ = ∅)
13	6, 11, 12	ifbieq12d 4529	. . 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 2782	. 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 4846	. . 3 ⊢ ⟨𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
16	15	csbeq2i 3882	. 2 ⊢ ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⦋𝐴 / 𝑥⦌if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17		dfopif 4846	. 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 2795	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459  [wsbc 3765  ⦋csb 3874  ∅c0 4308  ifcif 4500  {csn 4601  {cpr 4603  ⟨cop 4607

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608

This theorem is referenced by:  sbcop  5464  opsbc2ie  32457  esum2dlem  34123  csbfinxpg  37406