Theorem csbxpgVD 45430

Description: Virtual deduction proof of csbxp 5744. 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. csbxp 5744 is csbxpgVD 45430 without virtual deductions and was automatically derived from csbxpgVD 45430.

1::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑤 = 𝑤   )
4:3:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
5:2,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
6:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
7:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
8:7:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
9:6,8:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
10:5,9:	⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   )
11:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶))   )
12:10,11:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   )
13:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑧 = ⟨𝑤   ,    𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩)   )
14:12,13:	⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
15:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
16:14,15:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
17:16:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
18:17:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
19:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
20:18,19:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
21:20:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
22:21:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
23:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦 (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
24:22,23:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
25:24:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑧([𝐴 / 𝑥]∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
26:25:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}    )
27:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥] ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   )
28:26,27:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃ 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}    )
29::	⊢ {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
30::	⊢ (𝐵 × 𝐶) = {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
31:29,30:	⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤 , 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
32:31:	⊢ ∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
33:1,32:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   )
34:28,33:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}   )
35::	⊢ {⟨𝑤   ,   𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
36::	⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = { ⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)}
37:35,36:	⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
38:34,37:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)   )
qed:38:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbxpgVD	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem csbxpgVD

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 45111	. . . . . . . . . . . . . . . . . . 19 ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2		sbcel12 4362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)
3	2	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵))
4	1, 3	e1a 45164	. . . . . . . . . . . . . . . . . 18 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
5		csbconstg 3869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑤 = 𝑤)
6	1, 5	e1a 45164	. . . . . . . . . . . . . . . . . . 19 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑤 = 𝑤   )
7		eleq1 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (⦋𝐴 / 𝑥⦌𝑤 = 𝑤 → (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵))
8	6, 7	e1a 45164	. . . . . . . . . . . . . . . . . 18 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
9		bibi1 353	. . . . . . . . . . . . . . . . . . 19 ⊢ (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) → (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) ↔ (⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
10	9	biimprd 250	. . . . . . . . . . . . . . . . . 18 ⊢ (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) → ((⦋𝐴 / 𝑥⦌𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) → ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)))
11	4, 8, 10	e11 45225	. . . . . . . . . . . . . . . . 17 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
12		sbcel12 4362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)
13	12	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
14	1, 13	e1a 45164	. . . . . . . . . . . . . . . . . 18 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
15		csbconstg 3869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
16	1, 15	e1a 45164	. . . . . . . . . . . . . . . . . . 19 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌𝑦 = 𝑦   )
17		eleq1 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (⦋𝐴 / 𝑥⦌𝑦 = 𝑦 → (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
18	16, 17	e1a 45164	. . . . . . . . . . . . . . . . . 18 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
19		bibi1 353	. . . . . . . . . . . . . . . . . . 19 ⊢ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) → (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) ↔ (⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
20	19	biimprd 250	. . . . . . . . . . . . . . . . . 18 ⊢ (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) → ((⦋𝐴 / 𝑥⦌𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
21	14, 18, 20	e11 45225	. . . . . . . . . . . . . . . . 17 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)   )
22		pm4.38 646	. . . . . . . . . . . . . . . . . 18 ⊢ ((([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) ∧ ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) → (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
23	22	ex 416	. . . . . . . . . . . . . . . . 17 ⊢ (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵) → (([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶) → (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
24	11, 21, 23	e11 45225	. . . . . . . . . . . . . . . 16 ⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   )
25		sbcan 3791	. . . . . . . . . . . . . . . . . 18 ⊢ ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶))
26	25	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
27	1, 26	e1a 45164	. . . . . . . . . . . . . . . 16 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶))   )
28		bibi1 353	. . . . . . . . . . . . . . . . 17 ⊢ (([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) → (([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) ↔ (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
29	28	biimprcd 252	. . . . . . . . . . . . . . . 16 ⊢ ((([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) → (([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶)) → ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
30	24, 27, 29	e11 45225	. . . . . . . . . . . . . . 15 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))   )
31		sbcg 3814	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
32	1, 31	e1a 45164	. . . . . . . . . . . . . . 15 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩)   )
33		pm4.38 646	. . . . . . . . . . . . . . . 16 ⊢ ((([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩) ∧ ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
34	33	expcom 417	. . . . . . . . . . . . . . 15 ⊢ (([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)) → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩) → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))))
35	30, 32, 34	e11 45225	. . . . . . . . . . . . . 14 ⊢ (   𝐴 ∈ 𝑉   ▶   (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
36		sbcan 3791	. . . . . . . . . . . . . . . 16 ⊢ ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
37	36	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
38	1, 37	e1a 45164	. . . . . . . . . . . . . 14 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
39		bibi1 353	. . . . . . . . . . . . . . 15 ⊢ (([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) → (([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) ↔ (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))))
40	39	biimprcd 252	. . . . . . . . . . . . . 14 ⊢ ((([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) → (([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) → ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))))
41	35, 38, 40	e11 45225	. . . . . . . . . . . . 13 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
42	41	gen11 45153	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
43		exbi 1866	. . . . . . . . . . . 12 ⊢ (∀𝑦([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) → (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
44	42, 43	e1a 45164	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
45		sbcex2 3802	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
46	45	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
47	1, 46	e1a 45164	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
48		bibi1 353	. . . . . . . . . . . 12 ⊢ (([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) → (([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) ↔ (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))))
49	48	biimprcd 252	. . . . . . . . . . 11 ⊢ ((∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) → (([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) → ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))))
50	44, 47, 49	e11 45225	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
51	50	gen11 45153	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
52		exbi 1866	. . . . . . . . 9 ⊢ (∀𝑤([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) → (∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
53	51, 52	e1a 45164	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
54		sbcex2 3802	. . . . . . . . . 10 ⊢ ([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
55	54	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
56	1, 55	e1a 45164	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))   )
57		bibi1 353	. . . . . . . . 9 ⊢ (([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) → (([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) ↔ (∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))))
58	57	biimprcd 252	. . . . . . . 8 ⊢ ((∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) → (([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) → ([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))))
59	53, 56, 58	e11 45225	. . . . . . 7 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
60	59	gen11 45153	. . . . . 6 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑧([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))   )
61		abbib 2830	. . . . . . 7 ⊢ ({𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} ↔ ∀𝑧([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
62	61	biimpri 230	. . . . . 6 ⊢ (∀𝑧([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))) → {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))})
63	60, 62	e1a 45164	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}   )
64		csbab 4391	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
65	64	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))})
66	1, 65	e1a 45164	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   )
67		eqeq2 2773	. . . . . 6 ⊢ ({𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} → (⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} ↔ ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}))
68	67	biimpd 231	. . . . 5 ⊢ ({𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} → (⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}))
69	63, 66, 68	e11 45225	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}   )
70		df-xp 5649	. . . . . . 7 ⊢ (𝐵 × 𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
71		df-opab 5160	. . . . . . 7 ⊢ {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
72	70, 71	eqtri 2784	. . . . . 6 ⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
73	72	ax-gen 1814	. . . . 5 ⊢ ∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
74		csbeq2 3855	. . . . . 6 ⊢ (∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))})
75	74	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}))
76	1, 73, 75	e10 45231	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}   )
77		eqeq2 2773	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} → (⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} ↔ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}))
78	77	biimpd 231	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} → (⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}))
79	69, 76, 78	e11 45225	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}   )
80		df-xp 5649	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)}
81		df-opab 5160	. . . 4 ⊢ {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
82	80, 81	eqtri 2784	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
83		eqeq2 2773	. . . 4 ⊢ ((⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} → (⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}))
84	83	biimprcd 252	. . 3 ⊢ (⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} → ((⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))} → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)))
85	79, 82, 84	e10 45231	. 2 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)   )
86	85	in1 45108	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  [wsbc 3742  ⦋csb 3850  ⟨cop 4585  {copab 5159   × cxp 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-nul 4284  df-opab 5160  df-xp 5649  df-vd1 45107

This theorem is referenced by: (None)