Theorem csbingOLD 38076

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 3962 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem csbingOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3502	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷))
2		csbeq1 3502	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
3		csbeq1 3502	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
4	2, 3	ineq12d 3777	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
5	1, 4	eqeq12d 2625	. 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 3176	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3515	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
8		nfcsb1v 3515	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐷
9	7, 8	nfin 3782	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
10		csbeq1a 3508	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
11		csbeq1a 3508	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷)
12	10, 11	ineq12d 3777	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷))
13	6, 9, 12	csbief 3524	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
14	5, 13	vtoclg 3239	1 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ⦋csb 3499   ∩ cin 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-in 3547

This theorem is referenced by:  csbresgOLD  38077  onfrALTlem5VD  38143  onfrALTlem4VD  38144  csbresgVD  38153