Theorem csbif 4546

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 3865	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶))
2		dfsbcq2 3756	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		csbeq1 3865	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3865	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	ifbieq12d 4517	. . . 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 2745	. . 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 3451	. . . 4 ⊢ 𝑦 ∈ V
8		nfs1v 2157	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
9		nfcsb1v 3886	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
10		nfcsb1v 3886	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfif 4519	. . . 4 ⊢ Ⅎ𝑥if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
12		sbequ12 2252	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13		csbeq1a 3876	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3876	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	ifbieq12d 4517	. . . 4 ⊢ (𝑥 = 𝑦 → if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶))
16	7, 11, 15	csbief 3896	. . 3 ⊢ ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
17	6, 16	vtoclg 3520	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
18		csbprc 4372	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ∅)
19		csbprc 4372	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
20		csbprc 4372	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
21	19, 20	ifeq12d 4510	. . . 4 ⊢ (¬ 𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶) = if([𝐴 / 𝑥]𝜑, ∅, ∅))
22		ifid 4529	. . . 4 ⊢ if([𝐴 / 𝑥]𝜑, ∅, ∅) = ∅
23	21, 22	eqtr2di 2781	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
24	18, 23	eqtrd 2764	. 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 1540  [wsb 2065   ∈ wcel 2109  Vcvv 3447  [wsbc 3753  ⦋csb 3862  ∅c0 4296  ifcif 4488

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-nul 4297  df-if 4489

This theorem is referenced by:  csbopg  4855  fvmptnn04if  22736  csbrdgg  37317  csbfinxpg  37376  cdlemk40  40911