Theorem csbopg 4896

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 4588	. . 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 3844	. . . . 5 ⊢ ([𝐴 / 𝑥](𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ ([𝐴 / 𝑥]𝐶 ∈ V ∧ [𝐴 / 𝑥]𝐷 ∈ V))
3		sbcel1g 4422	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐶 ∈ V ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ V))
4		sbcel1g 4422	. . . . . 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 4714	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}})
8		csbsng 4713	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶} = {⦋𝐴 / 𝑥⦌𝐶})
9		csbprg 4714	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝐶, 𝐷} = {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷})
10	8, 9	preq12d 4746	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → {⦋𝐴 / 𝑥⦌{𝐶}, ⦋𝐴 / 𝑥⦌{𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
11	7, 10	eqtrd 2775	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{{𝐶}, {𝐶, 𝐷}} = {{⦋𝐴 / 𝑥⦌𝐶}, {⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷}})
12		csbconstg 3927	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∅ = ∅)
13	6, 11, 12	ifbieq12d 4559	. . 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 2787	. 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 4875	. . 3 ⊢ ⟨𝐶, 𝐷⟩ = if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
16	15	csbeq2i 3916	. 2 ⊢ ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⦋𝐴 / 𝑥⦌if((𝐶 ∈ V ∧ 𝐷 ∈ V), {{𝐶}, {𝐶, 𝐷}}, ∅)
17		dfopif 4875	. 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 2800	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝐶, 𝐷⟩ = ⟨⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  [wsbc 3791  ⦋csb 3908  ∅c0 4339  ifcif 4531  {csn 4631  {cpr 4633  ⟨cop 4637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638

This theorem is referenced by:  sbcop  5500  opsbc2ie  32504  esum2dlem  34073  csbfinxpg  37371