Theorem csbopg 4784

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 4480	. . 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 3747	. . . . 5 ⊢ ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3		sbcel1g 4313	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ V))
4		sbcel1g 4313	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐷 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐷 ∈ V))
5	3, 4	anbi12d 633	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V) ↔ (⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V)))
6	2, 5	syl5bb 286	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (⦋𝐴 / 𝑥⦌𝐶 ∈ V ∧ ⦋𝐴 / 𝑥⦌𝐷 ∈ V)))
7		csbprg 4605	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}})
8		csbsng 4604	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶} = {⦋𝐴 / 𝑥⦌𝐶})
9		csbprg 4605	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶, 𝐷} = {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷})
10	8, 9	preq12d 4637	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
11	7, 10	eqtrd 2793	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
12		csbconstg 3826	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∅ = ∅)
13	6, 11, 12	ifbieq12d 4451	. . 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 2805	. 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 4760	. . 3 ⊢ ⟨𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
16	15	csbeq2i 3815	. 2 ⊢ ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⦋𝐴 / 𝑥⦌if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17		dfopif 4760	. 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 2818	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  [wsbc 3698  ⦋csb 3807  ∅c0 4227  ifcif 4423  {csn 4525  {cpr 4527  ⟨cop 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532

This theorem is referenced by:  sbcop  5352  opsbc2ie  30359  esum2dlem  31592  csbfinxpg  35120