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Theorem csbresgVD 40789
Description: Virtual deduction proof of csbres 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. csbres 5744 is csbresgVD 40789 without virtual deductions and was automatically derived from csbresgVD 40789.
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 40468 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴𝑉   )
2 csbconstg 3834 . . . . . . . . 9 (𝐴𝑉𝐴 / 𝑥V = V)
31, 2e1a 40521 . . . . . . . 8 (   𝐴𝑉   ▶   𝐴 / 𝑥V = V   )
4 xpeq2 5471 . . . . . . . 8 (𝐴 / 𝑥V = V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
53, 4e1a 40521 . . . . . . 7 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V)   )
6 csbxp 5543 . . . . . . . . 9 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
76a1i 11 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V))
81, 7e1a 40521 . . . . . . 7 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)   )
9 eqeq2 2808 . . . . . . . 8 ((𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) ↔ 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)))
109biimpd 230 . . . . . . 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 40582 . . . . . 6 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)   )
12 ineq2 4109 . . . . . 6 (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
1311, 12e1a 40521 . . . . 5 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
14 csbin 4312 . . . . . . 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 40521 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))   )
17 eqeq2 2808 . . . . . 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 230 . . . . 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 40582 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
20 df-res 5462 . . . . . 6 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
2120ax-gen 1781 . . . . 5 𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
22 csbeq2 3822 . . . . . 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 40588 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))   )
25 eqeq2 2808 . . . . 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 230 . . . 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 40582 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
28 df-res 5462 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
29 eqeq2 2808 . . . 4 ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
3029biimprcd 251 . . 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 40588 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)   )
3231in1 40465 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1523   = wceq 1525  wcel 2083  Vcvv 3440  csb 3817  cin 3864   × cxp 5448  cres 5452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-in 3872  df-nul 4218  df-opab 5031  df-xp 5456  df-res 5462  df-vd1 40464
This theorem is referenced by: (None)
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