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Theorem csbresgVD 39445
Description: Virtual deduction proof of csbresgOLD 39370. 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. csbresgOLD 39370 is csbresgVD 39445 without virtual deductions and was automatically derived from csbresgVD 39445.
 1:: ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   ) 2:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌V = V   ) 3:2: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   ) 4:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)   ) 5:3,4: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   ) 6:5: ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   ) 7:1: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))   ) 8:6,7: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   ) 9:: ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) 10:9: ⊢ ∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) 11:1,10: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))   ) 12:8,11: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   ) 13:: ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) 14:12,13: ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)   ) qed:14: ⊢ (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
csbresgVD (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem csbresgVD
StepHypRef Expression
1 idn1 39107 . . . . . . . . 9 (   𝐴𝑉   ▶   𝐴𝑉   )
2 csbconstg 3579 . . . . . . . . 9 (𝐴𝑉𝐴 / 𝑥V = V)
31, 2e1a 39169 . . . . . . . 8 (   𝐴𝑉   ▶   𝐴 / 𝑥V = V   )
4 xpeq2 5163 . . . . . . . 8 (𝐴 / 𝑥V = V → (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V))
53, 4e1a 39169 . . . . . . 7 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V)   )
6 csbxpgOLD 39368 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V))
71, 6e1a 39169 . . . . . . 7 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V)   )
8 eqeq2 2662 . . . . . . . 8 ((𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) ↔ 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)))
98biimpd 219 . . . . . . 7 ((𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × 𝐴 / 𝑥V) → 𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)))
105, 7, 9e11 39230 . . . . . 6 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V)   )
11 ineq2 3841 . . . . . 6 (𝐴 / 𝑥(𝐶 × V) = (𝐴 / 𝑥𝐶 × V) → (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)))
1210, 11e1a 39169 . . . . 5 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
13 csbingOLD 39369 . . . . . 6 (𝐴𝑉𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V)))
141, 13e1a 39169 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵𝐴 / 𝑥(𝐶 × V))   )
15 eqeq2 2662 . . . . . 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))))
1615biimpd 219 . . . . 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))))
1712, 14, 16e11 39230 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵 ∩ (𝐶 × V)) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
18 df-res 5155 . . . . . 6 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
1918ax-gen 1762 . . . . 5 𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
20 csbeq2gOLD 39082 . . . . 5 (𝐴𝑉 → (∀𝑥(𝐵𝐶) = (𝐵 ∩ (𝐶 × V)) → 𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))))
211, 19, 20e10 39236 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = 𝐴 / 𝑥(𝐵 ∩ (𝐶 × V))   )
22 eqeq2 2662 . . . . 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))))
2322biimpd 219 . . . 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))))
2417, 21, 23e11 39230 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))   )
25 df-res 5155 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))
26 eqeq2 2662 . . . 4 ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → (𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ↔ 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V))))
2726biimprcd 240 . . 3 (𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → ((𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) = (𝐴 / 𝑥𝐵 ∩ (𝐴 / 𝑥𝐶 × V)) → 𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)))
2824, 25, 27e10 39236 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)   )
2928in1 39104 1 (𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⦋csb 3566   ∩ cin 3606   × cxp 5141   ↾ cres 5145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-in 3614  df-nul 3949  df-opab 4746  df-xp 5149  df-res 5155  df-vd1 39103 This theorem is referenced by: (None)
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