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Theorem onfrALTlem4VD 43949
Description: Virtual deduction proof of onfrALTlem4 43606. 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. onfrALTlem4 43606 is onfrALTlem4VD 43949 without virtual deductions and was automatically derived from onfrALTlem4VD 43949.
1:: 𝑦 ∈ V
2:1: ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅)
3:1: 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥 𝑎𝑦 / 𝑥𝑥)
4:1: 𝑦 / 𝑥𝑎 = 𝑎
5:1: 𝑦 / 𝑥𝑥 = 𝑦
6:4,5: (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = ( 𝑎𝑦)
7:3,6: 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
8:1: 𝑦 / 𝑥∅ = ∅
9:7,8: (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥 ∅ ↔ (𝑎𝑦) = ∅)
10:2,9: ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎 𝑦) = ∅)
11:1: ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
12:11,10: (([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥]( 𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
13:1: ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
qed:13,12: ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem4VD ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
Distinct variable group:   𝑥,𝑎

Proof of Theorem onfrALTlem4VD
StepHypRef Expression
1 sbcan 3828 . 2 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
2 sbcel1v 3847 . . 3 ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
3 sbceqg 4408 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅))
43elv 3478 . . . 4 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅)
5 csbin 4438 . . . . . 6 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥)
6 csbconstg 3911 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑎 = 𝑎)
76elv 3478 . . . . . . 7 𝑦 / 𝑥𝑎 = 𝑎
8 vex 3476 . . . . . . . 8 𝑦 ∈ V
98csbvargi 4431 . . . . . . 7 𝑦 / 𝑥𝑥 = 𝑦
107, 9ineq12i 4209 . . . . . 6 (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = (𝑎𝑦)
115, 10eqtri 2758 . . . . 5 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
12 csb0 4406 . . . . 5 𝑦 / 𝑥∅ = ∅
1311, 12eqeq12i 2748 . . . 4 (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅ ↔ (𝑎𝑦) = ∅)
144, 13bitri 274 . . 3 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎𝑦) = ∅)
152, 14anbi12i 625 . 2 (([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
161, 15bitri 274 1 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1539  Vcvv 3472  [wsbc 3776  csb 3892  cin 3946  c0 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-in 3954  df-nul 4322
This theorem is referenced by: (None)
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