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Theorem csbingVD 44909
Description: Virtual deduction proof of csbin 4441. 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 4441 is csbingVD 44909 without virtual deductions and was automatically derived from csbingVD 44909.
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 44599 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 df-in 3957 . . . . . . . 8 (𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}
32ax-gen 1794 . . . . . . 7 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}
4 spsbc 3800 . . . . . . 7 (𝐴𝐵 → (∀𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} → [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
51, 3, 4e10 44719 . . . . . 6 (   𝐴𝐵   ▶   [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
6 sbceqg 4411 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} ↔ 𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
76biimpd 229 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
81, 5, 7e11 44713 . . . . 5 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
9 csbab 4439 . . . . . . 7 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}
109a1i 11 . . . . . 6 (𝐴𝐵𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)})
111, 10e1a 44652 . . . . 5 (   𝐴𝐵   ▶   𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
12 eqeq1 2740 . . . . . 6 (𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} ↔ 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}))
1312biimprd 248 . . . . 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 44713 . . . 4 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
15 sbcan 3837 . . . . . . . . 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 44652 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
18 sbcel2 4417 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
1918a1i 11 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
201, 19e1a 44652 . . . . . . . 8 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)   )
21 sbcel2 4417 . . . . . . . . . 10 ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)
2221a1i 11 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
231, 22e1a 44652 . . . . . . . 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 412 . . . . . . . 8 (([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) → (([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷) → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2620, 23, 25e11 44713 . . . . . . 7 (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
27 bibi1 351 . . . . . . . 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 248 . . . . . . 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 44713 . . . . . 6 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
3029gen11 44641 . . . . 5 (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
31 abbib 2810 . . . . . 6 ({𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} ↔ ∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
3231biimpri 228 . . . . 5 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)})
3330, 32e1a 44652 . . . 4 (   𝐴𝐵   ▶   {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
34 eqeq1 2740 . . . . 5 (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} ↔ {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3534biimprd 248 . . . 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 44713 . . 3 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
37 df-in 3957 . . 3 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}
38 eqeq2 2748 . . . 4 ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3938biimprcd 250 . . 3 (𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
4036, 37, 39e10 44719 . 2 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
4140in1 44596 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wcel 2107  {cab 2713  [wsbc 3787  csb 3898  cin 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-in 3957  df-nul 4333  df-vd1 44595
This theorem is referenced by: (None)
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