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Theorem csbingVD 41577
Description: Virtual deduction proof of csbin 4350. 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. csbin 4350 is csbingVD 41577 without virtual deductions and was automatically derived from csbingVD 41577.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷) }
20:2: 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦 𝐷)}
30:1,20: (   𝐴𝐵   ▶   [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
3:1,30: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
4:1: (   𝐴𝐵   ▶   𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶 𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
5:3,4: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
6:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦 𝐴 / 𝑥𝐶)   )
7:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦 𝐴 / 𝑥𝐷)   )
8:6,7: (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶 [𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷 ))   )
9:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
10:9,8: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
11:10: (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦 𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
12:11: (   𝐴𝐵   ▶   {𝑦[𝐴 / 𝑥](𝑦𝐶 𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
13:5,12: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
14:: (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = { 𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}
15:13,14: (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
qed:15: (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = ( 𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbingVD (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

Proof of Theorem csbingVD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 idn1 41267 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 df-in 3891 . . . . . . . 8 (𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}
32ax-gen 1797 . . . . . . 7 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}
4 spsbc 3736 . . . . . . 7 (𝐴𝐵 → (∀𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} → [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
51, 3, 4e10 41387 . . . . . 6 (   𝐴𝐵   ▶   [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
6 sbceqg 4320 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} ↔ 𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
76biimpd 232 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
81, 5, 7e11 41381 . . . . 5 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
9 csbab 4348 . . . . . . 7 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}
109a1i 11 . . . . . 6 (𝐴𝐵𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)})
111, 10e1a 41320 . . . . 5 (   𝐴𝐵   ▶   𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
12 eqeq1 2805 . . . . . 6 (𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} ↔ 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}))
1312biimprd 251 . . . . 5 (𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}))
148, 11, 13e11 41381 . . . 4 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
15 sbcan 3771 . . . . . . . . 9 ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))
1615a1i 11 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)))
171, 16e1a 41320 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
18 sbcel2 4326 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
1918a1i 11 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
201, 19e1a 41320 . . . . . . . 8 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)   )
21 sbcel2 4326 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)
2221a1i 11 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
231, 22e1a 41320 . . . . . . . 8 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)   )
24 pm4.38 637 . . . . . . . . 9 ((([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) ∧ ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)) → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
2524ex 416 . . . . . . . 8 (([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) → (([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷) → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2620, 23, 25e11 41381 . . . . . . 7 (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
27 bibi1 355 . . . . . . . 8 (([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)) → (([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ↔ (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2827biimprd 251 . . . . . . 7 (([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)) → ((([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2917, 26, 28e11 41381 . . . . . 6 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
3029gen11 41309 . . . . 5 (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
31 abbi 2868 . . . . . 6 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ↔ {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)})
3231biimpi 219 . . . . 5 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)})
3330, 32e1a 41320 . . . 4 (   𝐴𝐵   ▶   {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
34 eqeq1 2805 . . . . 5 (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} ↔ {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3534biimprd 251 . . . 4 (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} → ({𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3614, 33, 35e11 41381 . . 3 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
37 df-in 3891 . . 3 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}
38 eqeq2 2813 . . . 4 ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3938biimprcd 253 . . 3 (𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
4036, 37, 39e10 41387 . 2 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
4140in1 41264 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2112  {cab 2779  [wsbc 3723  csb 3831  cin 3883
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-in 3891  df-nul 4247  df-vd1 41263
This theorem is referenced by: (None)
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