Theorem csbingOLD 39054

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 4010 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 3536	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷))
2		csbeq1 3536	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
3		csbeq1 3536	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
4	2, 3	ineq12d 3815	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))
5	1, 4	eqeq12d 2637	. 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 3203	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3549	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
8		nfcsb1v 3549	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐷
9	7, 8	nfin 3820	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
10		csbeq1a 3542	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
11		csbeq1a 3542	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐷 = ⦋𝑦 / 𝑥⦌𝐷)
12	10, 11	ineq12d 3815	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷))
13	6, 9, 12	csbief 3558	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝑦 / 𝑥⦌𝐶 ∩ ⦋𝑦 / 𝑥⦌𝐷)
14	5, 13	vtoclg 3266	1 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ⦋csb 3533   ∩ cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-in 3581

This theorem is referenced by:  csbresgOLD  39055  onfrALTlem5VD  39121  onfrALTlem4VD  39122  csbresgVD  39131