Theorem csbif 4588

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 3911	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶))
2		dfsbcq2 3794	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		csbeq1 3911	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3911	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	ifbieq12d 4559	. . . 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 2751	. . 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 3482	. . . 4 ⊢ 𝑦 ∈ V
8		nfs1v 2154	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
9		nfcsb1v 3933	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
10		nfcsb1v 3933	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfif 4561	. . . 4 ⊢ Ⅎ𝑥if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
12		sbequ12 2249	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13		csbeq1a 3922	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3922	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	ifbieq12d 4559	. . . 4 ⊢ (𝑥 = 𝑦 → if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶))
16	7, 11, 15	csbief 3943	. . 3 ⊢ ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
17	6, 16	vtoclg 3554	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
18		csbprc 4415	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ∅)
19		csbprc 4415	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
20		csbprc 4415	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
21	19, 20	ifeq12d 4552	. . . 4 ⊢ (¬ 𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶) = if([𝐴 / 𝑥]𝜑, ∅, ∅))
22		ifid 4571	. . . 4 ⊢ if([𝐴 / 𝑥]𝜑, ∅, ∅) = ∅
23	21, 22	eqtr2di 2792	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
24	18, 23	eqtrd 2775	. 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 1537  [wsb 2062   ∈ wcel 2106  Vcvv 3478  [wsbc 3791  ⦋csb 3908  ∅c0 4339  ifcif 4531

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-nul 4340  df-if 4532

This theorem is referenced by:  csbopg  4896  fvmptnn04if  22871  csbrdgg  37312  csbfinxpg  37371  cdlemk40  40900