Theorem csbif 4586

Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)

Assertion

Ref	Expression
csbif	⊢ ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbif

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3897	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶))
2		dfsbcq2 3781	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		csbeq1 3897	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3897	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	ifbieq12d 4557	. . . 4 ⊢ (𝑦 = 𝐴 → if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
6	1, 5	eqeq12d 2749	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶)))
7		vex 3479	. . . 4 ⊢ 𝑦 ∈ V
8		nfs1v 2154	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
9		nfcsb1v 3919	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
10		nfcsb1v 3919	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfif 4559	. . . 4 ⊢ Ⅎ𝑥if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
12		sbequ12 2244	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13		csbeq1a 3908	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3908	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	ifbieq12d 4557	. . . 4 ⊢ (𝑥 = 𝑦 → if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶))
16	7, 11, 15	csbief 3929	. . 3 ⊢ ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
17	6, 16	vtoclg 3557	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
18		csbprc 4407	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ∅)
19		csbprc 4407	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
20		csbprc 4407	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
21	19, 20	ifeq12d 4550	. . . 4 ⊢ (¬ 𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶) = if([𝐴 / 𝑥]𝜑, ∅, ∅))
22		ifid 4569	. . . 4 ⊢ if([𝐴 / 𝑥]𝜑, ∅, ∅) = ∅
23	21, 22	eqtr2di 2790	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
24	18, 23	eqtrd 2773	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
25	17, 24	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1542  [wsb 2068   ∈ wcel 2107  Vcvv 3475  [wsbc 3778  ⦋csb 3894  ∅c0 4323  ifcif 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-nul 4324  df-if 4530

This theorem is referenced by:  csbopg  4892  fvmptnn04if  22351  csbrdgg  36210  csbfinxpg  36269  cdlemk40  39788