Theorem csbin 4461

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 3918	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐵 ∩ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶))
2		csbeq1 3918	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3918	. . . . 5 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
4	2, 3	ineq12d 4236	. . . 4 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
5	1, 4	eqeq12d 2750	. . 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 3486	. . . 4 ⊢ 𝑦 ∈ V
7		nfcsb1v 3940	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
8		nfcsb1v 3940	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	7, 8	nfin 4239	. . . 4 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶)
10		csbeq1a 3929	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3929	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
12	10, 11	ineq12d 4236	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∩ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶))
13	6, 9, 12	csbief 3950	. . 3 ⊢ ⦋𝑦 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑦 / 𝑥⦌𝐶)
14	5, 13	vtoclg 3561	. 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
15		csbprc 4428	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = ∅)
16		csbprc 4428	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
17		csbprc 4428	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
18	16, 17	ineq12d 4236	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶) = (∅ ∩ ∅))
19		in0 4414	. . . 4 ⊢ (∅ ∩ ∅) = ∅
20	18, 19	eqtr2di 2791	. . 3 ⊢ (¬ 𝐴 ∈ V → ∅ = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
21	15, 20	eqtrd 2774	. 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶))
22	14, 21	pm2.61i 182	1 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2103  Vcvv 3482  ⦋csb 3915   ∩ cin 3969  ∅c0 4347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-in 3977  df-nul 4348

This theorem is referenced by:  csbres  6011  csbpredg  6337  disjxpin  32601  onfrALTlem5  44453  onfrALTlem4  44454  onfrALTlem5VD  44796  onfrALTlem4VD  44797  csbresgVD  44806  disjinfi  45033