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Theorem csbresgVD 39880
Description: Virtual deduction proof of csbres 5602. 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 5602 is csbresgVD 39880 without virtual deductions and was automatically derived from csbresgVD 39880.
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 39549 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴𝑉   )
2 csbconstg 3740 . . . . . . . . 9 (𝐴𝑉𝐴 / 𝑥V = V)
31, 2e1a 39611 . . . . . . . 8 (   𝐴𝑉   ▶   𝐴 / 𝑥V = V   )
4 xpeq2 5332 . . . . . . . 8 (𝐴 / 𝑥V = V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
53, 4e1a 39611 . . . . . . 7 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V)   )
6 csbxp 5404 . . . . . . . . 9 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)
76a1i 11 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V))
81, 7e1a 39611 . . . . . . 7 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)   )
9 eqeq2 2809 . . . . . . . 8 ((𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) ↔ 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)))
109biimpd 221 . . . . . . 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 39672 . . . . . 6 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)   )
12 ineq2 4005 . . . . . 6 (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
1311, 12e1a 39611 . . . . 5 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
14 csbin 4205 . . . . . . 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 39611 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))   )
17 eqeq2 2809 . . . . . 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 221 . . . . 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 39672 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
20 df-res 5323 . . . . . 6 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
2120ax-gen 1891 . . . . 5 𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
22 csbeq2 3731 . . . . . 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 39678 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))   )
25 eqeq2 2809 . . . . 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 221 . . . 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 39672 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
28 df-res 5323 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
29 eqeq2 2809 . . . 4 ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
3029biimprcd 242 . . 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 39678 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)   )
3231in1 39546 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651   = wceq 1653  wcel 2157  Vcvv 3384  csb 3727  cin 3767   × cxp 5309  cres 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-in 3775  df-nul 4115  df-opab 4905  df-xp 5317  df-res 5323  df-vd1 39545
This theorem is referenced by: (None)
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