Theorem csbif 4529

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 3845	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶))
2		dfsbcq2 3729	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		csbeq1 3845	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4		csbeq1 3845	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	ifbieq12d 4500	. . . 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 2752	. . 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 3445	. . . 4 ⊢ 𝑦 ∈ V
8		nfs1v 2152	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
9		nfcsb1v 3867	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
10		nfcsb1v 3867	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
11	8, 9, 10	nfif 4502	. . . 4 ⊢ Ⅎ𝑥if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
12		sbequ12 2243	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
13		csbeq1a 3856	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
14		csbeq1a 3856	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
15	12, 13, 14	ifbieq12d 4500	. . . 4 ⊢ (𝑥 = 𝑦 → if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶))
16	7, 11, 15	csbief 3877	. . 3 ⊢ ⦋𝑦 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝑦 / 𝑥]𝜑, ⦋𝑦 / 𝑥⦌𝐵, ⦋𝑦 / 𝑥⦌𝐶)
17	6, 16	vtoclg 3514	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
18		csbprc 4352	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌if(𝜑, 𝐵, 𝐶) = ∅)
19		csbprc 4352	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
20		csbprc 4352	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
21	19, 20	ifeq12d 4493	. . . 4 ⊢ (¬ 𝐴 ∈ V → if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶) = if([𝐴 / 𝑥]𝜑, ∅, ∅))
22		ifid 4512	. . . 4 ⊢ if([𝐴 / 𝑥]𝜑, ∅, ∅) = ∅
23	21, 22	eqtr2di 2793	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = if([𝐴 / 𝑥]𝜑, ⦋𝐴 / 𝑥⦌𝐵, ⦋𝐴 / 𝑥⦌𝐶))
24	18, 23	eqtrd 2776	. 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 2066   ∈ wcel 2105  Vcvv 3441  [wsbc 3726  ⦋csb 3842  ∅c0 4268  ifcif 4472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-nul 4269  df-if 4473

This theorem is referenced by:  csbopg  4834  fvmptnn04if  22096  csbrdgg  35598  csbfinxpg  35657  cdlemk40  39178