Theorem csbin 4399

Description: Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
csbin	⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem csbin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3849	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∩ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶))
2		csbeq1 3849	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3849	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	ineq12d 4166	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2776	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶)))
6		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3870	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3870	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfin 4169	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3860	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3860	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	ineq12d 4166	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∩ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3880	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶)
14	5, 13	vtoclg 3517	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
15		csbprc 4366	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = ∅)
16		csbprc 4366	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
17		csbprc 4366	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
18	16, 17	ineq12d 4166	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∩ ∅))
19		in0 4344	. . . 4 ⊢ (∅ ∩ ∅) = ∅
20	18, 19	eqtr2di 2812	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
21	15, 20	eqtrd 2795	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
22	14, 21	pm2.61i 184	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⦋csb 3846   ∩ cin 3897  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-in 3905  df-nul 4279

This theorem is used by:  csbres  5969  csbpredg  6299  disjxpin  33115  onfrALTlem5  45469  onfrALTlem4  45470  onfrALTlem5VD  45811  onfrALTlem4VD  45812  csbresgVD  45821  disjinfi  46128