Theorem csbingVD 42504

Description: Virtual deduction proof of csbin 4373. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbin 4373 is csbingVD 42504 without virtual deductions and was automatically derived from csbingVD 42504.

1::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) }
20:2:	⊢ ∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
30:1,20:	⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
3:1,30:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
4:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
5:3,4:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
6:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
7:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   )
8:6,7:	⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷 ))   )
9:1:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷))   )
10:9,8:	⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
11:10:	⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
12:11:	⊢ (   𝐴 ∈ 𝐵   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
13:5,12:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
14::	⊢ (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = { 𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}
15:13,14:	⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)   )
qed:15:	⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ( ⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem csbingVD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 42194	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		df-in 3894	. . . . . . . 8 ⊢ (𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
3	2	ax-gen 1798	. . . . . . 7 ⊢ ∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
4		spsbc 3729	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} → [𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}))
5	1, 3, 4	e10 42314	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
6		sbceqg 4343	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} ↔ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}))
7	6	biimpd 228	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}))
8	1, 5, 7	e11 42308	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
9		csbab 4371	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
10	9	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)})
11	1, 10	e1a 42247	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
12		eqeq1 2742	. . . . . 6 ⊢ (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} → (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} ↔ ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}))
13	12	biimprd 247	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} → (⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}))
14	8, 11, 13	e11 42308	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
15		sbcan 3768	. . . . . . . . 9 ⊢ ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷))
16	15	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷)))
17	1, 16	e1a 42247	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷))   )
18		sbcel2 4349	. . . . . . . . . 10 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)
19	18	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
20	1, 19	e1a 42247	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
21		sbcel2 4349	. . . . . . . . . 10 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)
22	21	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))
23	1, 22	e1a 42247	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   )
24		pm4.38 635	. . . . . . . . 9 ⊢ ((([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) ∧ ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)))
25	24	ex 413	. . . . . . . 8 ⊢ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) → (([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷) → (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
26	20, 23, 25	e11 42308	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
27		bibi1 352	. . . . . . . 8 ⊢ (([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷)) → (([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) ↔ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
28	27	biimprd 247	. . . . . . 7 ⊢ (([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷)) → ((([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))))
29	17, 26, 28	e11 42308	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
30	29	gen11 42236	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
31		abbi 2810	. . . . . 6 ⊢ (∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) ↔ {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)})
32	31	biimpi 215	. . . . 5 ⊢ (∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)) → {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)})
33	30, 32	e1a 42247	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
34		eqeq1 2742	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} → (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} ↔ {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}))
35	34	biimprd 247	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} → ({𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}))
36	14, 33, 35	e11 42308	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
37		df-in 3894	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}
38		eqeq2 2750	. . . 4 ⊢ ((⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} → (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) ↔ ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}))
39	38	biimprcd 249	. . 3 ⊢ (⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} → ((⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)))
40	36, 37, 39	e10 42314	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)   )
41	40	in1 42191	1 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  [wsbc 3716  ⦋csb 3832   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-in 3894  df-nul 4257  df-vd1 42190

This theorem is referenced by: (None)