Theorem csbingVD 45000

Description: Virtual deduction proof of csbin 4391. 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 4391 is csbingVD 45000 without virtual deductions and was automatically derived from csbingVD 45000.

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 44691	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2		df-in 3905	. . . . . . . 8 ⊢ (𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
3	2	ax-gen 1796	. . . . . . 7 ⊢ ∀𝑥(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}
4		spsbc 3750	. . . . . . 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 44811	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
6		sbceqg 4361	. . . . . . 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 229	. . . . . 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 44805	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
9		csbab 4389	. . . . . . 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 44744	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
12		eqeq1 2737	. . . . . 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 248	. . . . 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 44805	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}   )
15		sbcan 3787	. . . . . . . . 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 44744	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷))   )
18		sbcel2 4367	. . . . . . . . . 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 44744	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
21		sbcel2 4367	. . . . . . . . . 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 44744	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐷 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)   )
24		pm4.38 637	. . . . . . . . 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 412	. . . . . . . 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 44805	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
27		bibi1 351	. . . . . . . 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 248	. . . . . . 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 44805	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
30	29	gen11 44733	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ▶   ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷))   )
31		abbib 2802	. . . . . 6 ⊢ ({𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)} ↔ ∀𝑦([𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)))
32	31	biimpri 228	. . . . 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 44744	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ▶   {𝑦 ∣ [𝐴 / 𝑥](𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
34		eqeq1 2737	. . . . 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 248	. . . 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 44805	. . 3 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}   )
37		df-in 3905	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷) = {𝑦 ∣ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐷)}
38		eqeq2 2745	. . . 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 250	. . 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 44811	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)   )
41	40	in1 44688	1 ⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2711  [wsbc 3737  ⦋csb 3846   ∩ cin 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-in 3905  df-nul 4283  df-vd1 44687

This theorem is referenced by: (None)