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Theorem csbresgVD 43167
Description: Virtual deduction proof of csbres 5940. 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. csbres 5940 is csbresgVD 43167 without virtual deductions and was automatically derived from csbresgVD 43167.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   𝐴 / 𝑥V = V   )
3:2: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V)   )
4:1: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)   )
5:3,4: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)   )
6:5: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
7:1: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))   )
8:6,7: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
9:: (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
10:9: 𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
11:1,10: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))   )
12:8,11: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = ( 𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
13:: (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = ( 𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
14:12,13: (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = ( 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)   )
qed:14: (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = ( 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbresgVD (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem csbresgVD
StepHypRef Expression
1 idn1 42846 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴𝑉   )
2 csbconstg 3874 . . . . . . . . 9 (𝐴𝑉𝐴 / 𝑥V = V)
31, 2e1a 42899 . . . . . . . 8 (   𝐴𝑉   ▶   𝐴 / 𝑥V = V   )
4 xpeq2 5654 . . . . . . . 8 (𝐴 / 𝑥V = V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
53, 4e1a 42899 . . . . . . 7 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V)   )
6 csbxp 5731 . . . . . . . . 9 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
76a1i 11 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V))
81, 7e1a 42899 . . . . . . 7 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)   )
9 eqeq2 2748 . . . . . . . 8 ((𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) ↔ 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)))
109biimpd 228 . . . . . . 7 ((𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)))
115, 8, 10e11 42960 . . . . . 6 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)   )
12 ineq2 4166 . . . . . 6 (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
1311, 12e1a 42899 . . . . 5 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
14 csbin 4399 . . . . . . 7 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))
1514a1i 11 . . . . . 6 (𝐴𝑉𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)))
161, 15e1a 42899 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))   )
17 eqeq2 2748 . . . . . 6 ((𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) ↔ 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
1817biimpd 228 . . . . 5 ((𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) → 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
1913, 16, 18e11 42960 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
20 df-res 5645 . . . . . 6 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
2120ax-gen 1797 . . . . 5 𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
22 csbeq2 3860 . . . . . 6 (∀𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V)) → 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)))
2322a1i 11 . . . . 5 (𝐴𝑉 → (∀𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V)) → 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))))
241, 21, 23e10 42966 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))   )
25 eqeq2 2748 . . . . 5 (𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
2625biimpd 228 . . . 4 (𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
2719, 24, 26e11 42960 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
28 df-res 5645 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
29 eqeq2 2748 . . . 4 ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
3029biimprcd 249 . . 3 (𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
3127, 28, 30e10 42966 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)   )
3231in1 42843 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539   = wceq 1541  wcel 2106  Vcvv 3445  csb 3855  cin 3909   × cxp 5631  cres 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-in 3917  df-nul 4283  df-opab 5168  df-xp 5639  df-res 5645  df-vd1 42842
This theorem is referenced by: (None)
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