Theorem csbin 4406

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 3855	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∩ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶))
2		csbeq1 3855	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3855	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	ineq12d 4173	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2778	. . 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 3458	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3876	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3876	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfin 4176	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3866	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3866	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	ineq12d 4173	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∩ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3886	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶)
14	5, 13	vtoclg 3521	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
15		csbprc 4373	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = ∅)
16		csbprc 4373	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
17		csbprc 4373	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
18	16, 17	ineq12d 4173	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∩ ∅))
19		in0 4351	. . . 4 ⊢ (∅ ∩ ∅) = ∅
20	18, 19	eqtr2di 2814	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
21	15, 20	eqtrd 2797	. 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 1569   ∈ wcel 2142  Vcvv 3454  ⦋csb 3852   ∩ cin 3903  ∅c0 4285

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-in 3911  df-nul 4286

This theorem is used by:  csbres  5980  csbpredg  6308  disjxpin  32944  onfrALTlem5  45279  onfrALTlem4  45280  onfrALTlem5VD  45621  onfrALTlem4VD  45622  csbresgVD  45631  disjinfi  45938