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Theorem sbcssgVD 39641
Description: Virtual deduction proof of sbcssg 4224. 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 4224 is sbcssgVD 39641 without virtual deductions and was automatically derived from sbcssgVD 39641.
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 39315 . . . . . . . . . 10 (   𝐴𝐵   ▶   𝐴𝐵   )
2 sbcel2gOLD 39280 . . . . . . . . . 10 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶))
31, 2e1a 39377 . . . . . . . . 9 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶)   )
4 sbcel2gOLD 39280 . . . . . . . . . 10 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷))
51, 4e1a 39377 . . . . . . . . 9 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷)   )
6 imbi12 335 . . . . . . . . 9 (([𝐴 / 𝑥]𝑦𝐶𝑦𝐴 / 𝑥𝐶) → (([𝐴 / 𝑥]𝑦𝐷𝑦𝐴 / 𝑥𝐷) → (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
73, 5, 6e11 39438 . . . . . . . 8 (   𝐴𝐵   ▶   (([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
8 sbcimg 3629 . . . . . . . . 9 (𝐴𝐵 → ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷)))
91, 8e1a 39377 . . . . . . . 8 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ([𝐴 / 𝑥]𝑦𝐶[𝐴 / 𝑥]𝑦𝐷))   )
10 bibi1 340 . . . . . . . . 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))))
1110biimprcd 240 . . . . . . . 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))))
127, 9, 11e11 39438 . . . . . . 7 (   𝐴𝐵   ▶   ([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
1312gen11 39366 . . . . . 6 (   𝐴𝐵   ▶   𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
14 albi 1894 . . . . . 6 (∀𝑦([𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ (𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)))
1513, 14e1a 39377 . . . . 5 (   𝐴𝐵   ▶   (∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
16 sbcalgOLD 39277 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷)))
171, 16e1a 39377 . . . . 5 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦[𝐴 / 𝑥](𝑦𝐶𝑦𝐷))   )
18 bibi1 340 . . . . . 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))))
1918biimprcd 240 . . . . 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))))
2015, 17, 19e11 39438 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
21 dfss2 3740 . . . . . 6 (𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
2221ax-gen 1870 . . . . 5 𝑥(𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷))
23 sbcbi 39274 . . . . 5 (𝐴𝐵 → (∀𝑥(𝐶𝐷 ↔ ∀𝑦(𝑦𝐶𝑦𝐷)) → ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷))))
241, 22, 23e10 39444 . . . 4 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷))   )
25 bibi1 340 . . . . 5 (([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷)) → (([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ↔ ([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2625biimprcd 240 . . . 4 (([𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷) ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → (([𝐴 / 𝑥]𝐶𝐷[𝐴 / 𝑥]𝑦(𝑦𝐶𝑦𝐷)) → ([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))))
2720, 24, 26e11 39438 . . 3 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))   )
28 dfss2 3740 . . 3 (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))
29 biantr 807 . . . 4 ((([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) ∧ (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷))) → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
3029ex 397 . . 3 (([𝐴 / 𝑥]𝐶𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → ((𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷 ↔ ∀𝑦(𝑦𝐴 / 𝑥𝐶𝑦𝐴 / 𝑥𝐷)) → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)))
3127, 28, 30e10 39444 . 2 (   𝐴𝐵   ▶   ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷)   )
3231in1 39312 1 (𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wcel 2145  [wsbc 3587  csb 3682  wss 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-vd1 39311
This theorem is referenced by: (None)
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