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Theorem onfrALTlem1VD 41101
Description: Virtual deduction proof of onfrALTlem1 40759. 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. onfrALTlem1 40759 is onfrALTlem1VD 41101 without virtual deductions and was automatically derived from onfrALTlem1VD 41101.
1:: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   )
2:1: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅)   )
3:2: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅)    )
4:: ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅ ) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
5:4: 𝑦([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
6:5: (∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
7:3,6: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)   )
8:: (∃𝑦𝑎(𝑎𝑦) = ∅ ↔ ∃𝑦( 𝑦𝑎 ∧ (𝑎𝑦) = ∅))
qed:7,8: (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 (𝑎𝑥) = ∅)   ▶   𝑦𝑎(𝑎𝑦) = ∅   )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem1VD (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
Distinct variable group:   𝑥,𝑎,𝑦

Proof of Theorem onfrALTlem1VD
StepHypRef Expression
1 idn2 40824 . . . . 5 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   )
2 19.8a 2170 . . . . 5 ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅))
31, 2e2 40842 . . . 4 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅)   )
4 cbvexsv 40758 . . . . 5 (∃𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅))
54biimpi 217 . . . 4 (∃𝑥(𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅))
63, 5e2 40842 . . 3 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅)   )
7 sbsbc 3773 . . . . . 6 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ [𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅))
8 onfrALTlem4 40754 . . . . . 6 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
97, 8bitri 276 . . . . 5 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
109ax-gen 1787 . . . 4 𝑦([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
11 exbi 1838 . . . 4 (∀𝑦([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅)) → (∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)))
1210, 11e0a 40983 . . 3 (∃𝑦[𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
136, 12e2bi 40843 . 2 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅)   )
14 df-rex 3141 . 2 (∃𝑦𝑎 (𝑎𝑦) = ∅ ↔ ∃𝑦(𝑦𝑎 ∧ (𝑎𝑦) = ∅))
1513, 14e2bir 40844 1 (   (𝑎 ⊆ On ∧ 𝑎 ≠ ∅)   ,   (𝑥𝑎 ∧ (𝑎𝑥) = ∅)   ▶   𝑦𝑎 (𝑎𝑦) = ∅   )
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wal 1526   = wceq 1528  wex 1771  [wsb 2060  wcel 2105  wne 3013  wrex 3136  [wsbc 3769  cin 3932  wss 3933  c0 4288  Oncon0 6184  (   wvd2 40788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-in 3940  df-nul 4289  df-vd2 40789
This theorem is referenced by:  onfrALTVD  41102
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