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Theorem csbxpgVD 42187
Description: Virtual deduction proof of csbxp 5647. 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 5647 is csbxpgVD 42187 without virtual deductions and was automatically derived from csbxpgVD 42187.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝐵 𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵)   )
3:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑤 = 𝑤   )
4:3: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵𝑤𝐴 / 𝑥𝐵)   )
5:2,4: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝐵𝑤 𝐴 / 𝑥𝐵)   )
6:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐶 𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶)   )
7:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
8:7: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐶)   )
9:6,8: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦 𝐴 / 𝑥𝐶)   )
10:5,9: (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑤𝐵 [𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶))   )
11:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑤𝐵 𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶))   )
12:10,11: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑤𝐵 𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))   )
13:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧 = ⟨𝑤   ,    𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩)   )
14:12,13: (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦 ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
15:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦 [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)))   )
16:14,15: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤    ,   𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
17:16: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
18:17: (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
19:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))   )
20:18,19: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
21:20: (   𝐴𝑉   ▶   𝑤([𝐴 / 𝑥]𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
22:21: (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
23:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦 (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))   )
24:22,23: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
25:24: (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
26:25: (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}    )
27:1: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥] 𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}   )
28:26,27: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑤 𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦( 𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}    )
29:: {⟨𝑤   ,   𝑦⟩ ∣ (𝑤𝐵𝑦𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
30:: (𝐵 × 𝐶) = {⟨𝑤   ,   𝑦⟩ ∣ (𝑤𝐵 𝑦𝐶)}
31:29,30: (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤 , 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
32:31: 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = 𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
33:1,32: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵 𝑦𝐶))}   )
34:28,33: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶))}   )
35:: {⟨𝑤   ,   𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}
36:: (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = { 𝑤, 𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)}
37:35,36: (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵 𝑦𝐴 / 𝑥𝐶))}
38:34,37: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶)   )
qed:38: (𝐴𝑉𝐴 / 𝑥(𝐵 × 𝐶) = ( 𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbxpgVD (𝐴𝑉𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))

Proof of Theorem csbxpgVD
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 41867 . . . . . . . . . . . . . . . . . . 19 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbcel12 4323 . . . . . . . . . . . . . . . . . . . 20 ([𝐴 / 𝑥]𝑤𝐵𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵)
32a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝐵𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵))
41, 3e1a 41920 . . . . . . . . . . . . . . . . . 18 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝐵𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵)   )
5 csbconstg 3830 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥𝑤 = 𝑤)
61, 5e1a 41920 . . . . . . . . . . . . . . . . . . 19 (   𝐴𝑉   ▶   𝐴 / 𝑥𝑤 = 𝑤   )
7 eleq1 2825 . . . . . . . . . . . . . . . . . . 19 (𝐴 / 𝑥𝑤 = 𝑤 → (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵𝑤𝐴 / 𝑥𝐵))
86, 7e1a 41920 . . . . . . . . . . . . . . . . . 18 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵𝑤𝐴 / 𝑥𝐵)   )
9 bibi1 355 . . . . . . . . . . . . . . . . . . 19 (([𝐴 / 𝑥]𝑤𝐵𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵) → (([𝐴 / 𝑥]𝑤𝐵𝑤𝐴 / 𝑥𝐵) ↔ (𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵𝑤𝐴 / 𝑥𝐵)))
109biimprd 251 . . . . . . . . . . . . . . . . . 18 (([𝐴 / 𝑥]𝑤𝐵𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵) → ((𝐴 / 𝑥𝑤𝐴 / 𝑥𝐵𝑤𝐴 / 𝑥𝐵) → ([𝐴 / 𝑥]𝑤𝐵𝑤𝐴 / 𝑥𝐵)))
114, 8, 10e11 41981 . . . . . . . . . . . . . . . . 17 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝐵𝑤𝐴 / 𝑥𝐵)   )
12 sbcel12 4323 . . . . . . . . . . . . . . . . . . . 20 ([𝐴 / 𝑥]𝑦𝐶𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶)
1312a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐶𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶))
141, 13e1a 41920 . . . . . . . . . . . . . . . . . 18 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐶𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶)   )
15 csbconstg 3830 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
161, 15e1a 41920 . . . . . . . . . . . . . . . . . . 19 (   𝐴𝑉   ▶   𝐴 / 𝑥𝑦 = 𝑦   )
17 eleq1 2825 . . . . . . . . . . . . . . . . . . 19 (𝐴 / 𝑥𝑦 = 𝑦 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐶))
1816, 17e1a 41920 . . . . . . . . . . . . . . . . . 18 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐶)   )
19 bibi1 355 . . . . . . . . . . . . . . . . . . 19 (([𝐴 / 𝑥]𝑦𝐶𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶) → (([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) ↔ (𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐶)))
2019biimprd 251 . . . . . . . . . . . . . . . . . 18 (([𝐴 / 𝑥]𝑦𝐶𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶) → ((𝐴 / 𝑥𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐶) → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)))
2114, 18, 20e11 41981 . . . . . . . . . . . . . . . . 17 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)   )
22 pm4.38 638 . . . . . . . . . . . . . . . . . 18 ((([𝐴 / 𝑥]𝑤𝐵𝑤𝐴 / 𝑥𝐵) ∧ ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)) → (([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))
2322ex 416 . . . . . . . . . . . . . . . . 17 (([𝐴 / 𝑥]𝑤𝐵𝑤𝐴 / 𝑥𝐵) → (([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) → (([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
2411, 21, 23e11 41981 . . . . . . . . . . . . . . . 16 (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))   )
25 sbcan 3746 . . . . . . . . . . . . . . . . . 18 ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶))
2625a1i 11 . . . . . . . . . . . . . . . . 17 (𝐴𝑉 → ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶)))
271, 26e1a 41920 . . . . . . . . . . . . . . . 16 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶))   )
28 bibi1 355 . . . . . . . . . . . . . . . . 17 (([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶)) → (([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)) ↔ (([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
2928biimprcd 253 . . . . . . . . . . . . . . . 16 ((([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)) → (([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ ([𝐴 / 𝑥]𝑤𝐵[𝐴 / 𝑥]𝑦𝐶)) → ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
3024, 27, 29e11 41981 . . . . . . . . . . . . . . 15 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))   )
31 sbcg 3774 . . . . . . . . . . . . . . . 16 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
321, 31e1a 41920 . . . . . . . . . . . . . . 15 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩)   )
33 pm4.38 638 . . . . . . . . . . . . . . . 16 ((([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩) ∧ ([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
3433expcom 417 . . . . . . . . . . . . . . 15 (([𝐴 / 𝑥](𝑤𝐵𝑦𝐶) ↔ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)) → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩) → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))))
3530, 32, 34e11 41981 . . . . . . . . . . . . . 14 (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
36 sbcan 3746 . . . . . . . . . . . . . . . 16 ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)))
3736a1i 11 . . . . . . . . . . . . . . 15 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶))))
381, 37e1a 41920 . . . . . . . . . . . . . 14 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)))   )
39 bibi1 355 . . . . . . . . . . . . . . 15 (([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶))) → (([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) ↔ (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))))
4039biimprcd 253 . . . . . . . . . . . . . 14 ((([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) → (([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤𝐵𝑦𝐶))) → ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))))
4135, 38, 40e11 41981 . . . . . . . . . . . . 13 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
4241gen11 41909 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
43 exbi 1854 . . . . . . . . . . . 12 (∀𝑦([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) → (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
4442, 43e1a 41920 . . . . . . . . . . 11 (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
45 sbcex2 3760 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))
4645a1i 11 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))))
471, 46e1a 41920 . . . . . . . . . . 11 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))   )
48 bibi1 355 . . . . . . . . . . . 12 (([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))) → (([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) ↔ (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))))
4948biimprcd 253 . . . . . . . . . . 11 ((∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) → (([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))) → ([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))))
5044, 47, 49e11 41981 . . . . . . . . . 10 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
5150gen11 41909 . . . . . . . . 9 (   𝐴𝑉   ▶   𝑤([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
52 exbi 1854 . . . . . . . . 9 (∀𝑤([𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) → (∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))))
5351, 52e1a 41920 . . . . . . . 8 (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
54 sbcex2 3760 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))
5554a1i 11 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))))
561, 55e1a 41920 . . . . . . . 8 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)))   )
57 bibi1 355 . . . . . . . . 9 (([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))) → (([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) ↔ (∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))))
5857biimprcd 253 . . . . . . . 8 ((∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) → (([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))) → ([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))))
5953, 56, 58e11 41981 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
6059gen11 41909 . . . . . 6 (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)))   )
61 abbi 2810 . . . . . . 7 (∀𝑧([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) ↔ {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))})
6261biimpi 219 . . . . . 6 (∀𝑧([𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶)) ↔ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))) → {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))})
6360, 62e1a 41920 . . . . 5 (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}   )
64 csbab 4352 . . . . . . 7 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
6564a1i 11 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))})
661, 65e1a 41920 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}   )
67 eqeq2 2749 . . . . . 6 ({𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))} → (𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} ↔ 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}))
6867biimpd 232 . . . . 5 ({𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))} → (𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧[𝐴 / 𝑥]𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} → 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}))
6963, 66, 68e11 41981 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}   )
70 df-xp 5557 . . . . . . 7 (𝐵 × 𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐵𝑦𝐶)}
71 df-opab 5116 . . . . . . 7 {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐵𝑦𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
7270, 71eqtri 2765 . . . . . 6 (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
7372ax-gen 1803 . . . . 5 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}
74 csbeq2 3816 . . . . . 6 (∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} → 𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))})
7574a1i 11 . . . . 5 (𝐴𝑉 → (∀𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} → 𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}))
761, 73, 75e10 41987 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))}   )
77 eqeq2 2749 . . . . 5 (𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))} → (𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} ↔ 𝐴 / 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}))
7877biimpd 232 . . . 4 (𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))} → (𝐴 / 𝑥(𝐵 × 𝐶) = 𝐴 / 𝑥{𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐵𝑦𝐶))} → 𝐴 / 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}))
7969, 76, 78e11 41981 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}   )
80 df-xp 5557 . . . 4 (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)}
81 df-opab 5116 . . . 4 {⟨𝑤, 𝑦⟩ ∣ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶)} = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}
8280, 81eqtri 2765 . . 3 (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}
83 eqeq2 2749 . . . 4 ((𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))} → (𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))}))
8483biimprcd 253 . . 3 (𝐴 / 𝑥(𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))} → ((𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶) = {𝑧 ∣ ∃𝑤𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤𝐴 / 𝑥𝐵𝑦𝐴 / 𝑥𝐶))} → 𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶)))
8579, 82, 84e10 41987 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶)   )
8685in1 41864 1 (𝐴𝑉𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wex 1787  wcel 2110  {cab 2714  [wsbc 3694  csb 3811  cop 4547  {copab 5115   × cxp 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-nul 4238  df-opab 5116  df-xp 5557  df-vd1 41863
This theorem is referenced by: (None)
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