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Theorem sbcssgVD 44856
Description: Virtual deduction proof of sbcssg 4543. 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. sbcssg 4543 is sbcssgVD 44856 without virtual deductions and was automatically derived from sbcssgVD 44856.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦 𝐴 / 𝑥𝐶)   )
3:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦 𝐴 / 𝑥𝐷)   )
4:2,3: (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶 [𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷 ))   )
5:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
6:4,5: (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶 𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
7:6: (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦 𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
8:7: (   𝐴𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦 𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷) )   )
9:1: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦 𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷))   )
10:8,9: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦 𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷) )   )
11:: (𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
110:11: 𝑥(𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦 𝐷))
12:1,110: (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷 [𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷))   )
13:10,12: (   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: (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷 𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcssgVD (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

Proof of Theorem sbcssgVD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 idn1 44547 . . . . . . . . . 10 (   𝐴𝐵   ▶   𝐴𝐵   )
2 sbcel2 4441 . . . . . . . . . . 11 ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)
32a1i 11 . . . . . . . . . 10 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
41, 3e1a 44600 . . . . . . . . 9 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)   )
5 sbcel2 4441 . . . . . . . . . . 11 ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)
65a1i 11 . . . . . . . . . 10 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
71, 6e1a 44600 . . . . . . . . 9 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)   )
8 imbi12 346 . . . . . . . . 9 (([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) → (([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷) → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
94, 7, 8e11 44661 . . . . . . . 8 (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
10 sbcimg 3856 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)))
111, 10e1a 44600 . . . . . . . 8 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
12 bibi1 351 . . . . . . . . 9 (([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)) → (([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ↔ (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
1312biimprcd 250 . . . . . . . 8 ((([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → (([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)) → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
149, 11, 13e11 44661 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
1514gen11 44589 . . . . . 6 (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
16 albi 1816 . . . . . 6 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
1715, 16e1a 44600 . . . . 5 (   𝐴𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
18 sbcal 3868 . . . . . . 7 ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷))
1918a1i 11 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)))
201, 19e1a 44600 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷))   )
21 bibi1 351 . . . . . 6 (([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)) → (([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ↔ (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2221biimprcd 250 . . . . 5 ((∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → (([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)) → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2317, 20, 22e11 44661 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
24 df-ss 3993 . . . . . 6 (𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
2524ax-gen 1793 . . . . 5 𝑥(𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
26 sbcbi 44512 . . . . 5 (𝐴𝐵 → (∀𝑥(𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷)) → ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷))))
271, 25, 26e10 44667 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷))   )
28 bibi1 351 . . . . 5 (([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷)) → (([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ↔ ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2928biimprcd 250 . . . 4 (([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → (([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷)) → ([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
3023, 27, 29e11 44661 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
31 df-ss 3993 . . 3 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))
32 biantr 805 . . . 4 ((([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ∧ (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))) → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
3332ex 412 . . 3 (([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
3430, 31, 33e10 44667 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
3534in1 44544 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wcel 2108  [wsbc 3804  csb 3921  wss 3976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-ss 3993  df-nul 4353  df-vd1 44543
This theorem is referenced by: (None)
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