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Theorem csbingVD 38142
Description: Virtual deduction proof of csbingOLD 38076. 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. csbingOLD 38076 is csbingVD 38142 without virtual deductions and was automatically derived from csbingVD 38142.
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 37811 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
2 df-in 3547 . . . . . . . 8 (𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}
32ax-gen 1713 . . . . . . 7 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}
4 spsbc 3415 . . . . . . 7 (𝐴𝐵 → (∀𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} → [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
51, 3, 4e10 37940 . . . . . 6 (   𝐴𝐵   ▶   [𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
6 sbceqg 3936 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} ↔ 𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
76biimpd 218 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥](𝐶𝐷) = {𝑦 ∣ (𝑦𝐶𝑦𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}))
81, 5, 7e11 37934 . . . . 5 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)}   )
9 csbabgOLD 38072 . . . . . 6 (𝐴𝐵𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)})
101, 9e1a 37873 . . . . 5 (   𝐴𝐵   ▶   𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
11 eqeq1 2614 . . . . . 6 (𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} ↔ 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}))
1211biimprd 237 . . . . 5 (𝐴 / 𝑥(𝐶𝐷) = 𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥{𝑦 ∣ (𝑦𝐶𝑦𝐷)} = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}))
138, 10, 12e11 37934 . . . 4 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)}   )
14 sbcangOLD 37760 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)))
151, 14e1a 37873 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
16 sbcel2gOLD 37776 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
171, 16e1a 37873 . . . . . . . 8 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)   )
18 sbcel2gOLD 37776 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
191, 18e1a 37873 . . . . . . . 8 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)   )
20 pm4.38 912 . . . . . . . . 9 ((([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) ∧ ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)) → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
2120ex 449 . . . . . . . 8 (([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) → (([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷) → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2217, 19, 21e11 37934 . . . . . . 7 (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
23 bibi1 340 . . . . . . . 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))))
2423biimprd 237 . . . . . . 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))))
2515, 22, 24e11 37934 . . . . . 6 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
2625gen11 37862 . . . . 5 (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
27 abbi 2724 . . . . . 6 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ↔ {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)})
2827biimpi 205 . . . . 5 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)})
2926, 28e1a 37873 . . . 4 (   𝐴𝐵   ▶   {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
30 eqeq1 2614 . . . . 5 (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} ↔ {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3130biimprd 237 . . . 4 (𝐴 / 𝑥(𝐶𝐷) = {𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} → ({𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)} = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3213, 29, 31e11 37934 . . 3 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}   )
33 df-in 3547 . . 3 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}
34 eqeq2 2621 . . . 4 ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → (𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) ↔ 𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)}))
3534biimprcd 239 . . 3 (𝐴 / 𝑥(𝐶𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷) = {𝑦 ∣ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)} → 𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
3632, 33, 35e10 37940 . 2 (   𝐴𝐵   ▶   𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
3736in1 37808 1 (𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wcel 1977  {cab 2596  [wsbc 3402  csb 3499  cin 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500  df-in 3547  df-vd1 37807
This theorem is referenced by: (None)
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