Theorem csbing 3144

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Assertion

Ref	Expression
csbing	⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Proof of Theorem csbing

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 2855	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷))
2		csbeq1 2855	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
3		csbeq1 2855	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
4	2, 3	ineq12d 3139	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
5	1, 4	eqeq12d 2054	. 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷) ↔ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)))
6		vex 2560	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 2882	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
8		nfcsb1v 2882	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐷
9	7, 8	nfin 3143	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
10		csbeq1a 2860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
11		csbeq1a 2860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷)
12	10, 11	ineq12d 3139	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷))
13	6, 9, 12	csbief 2891	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
14	5, 13	vtoclg 2613	1 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ⦋csb 2852   ∩ cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-in 2924

This theorem is referenced by:  csbresg  4615