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Theorem onfrALTlem4VD 38602
Description: Virtual deduction proof of onfrALTlem4 38237. 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 38237 is onfrALTlem4VD 38602 without virtual deductions and was automatically derived from onfrALTlem4VD 38602.
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 vex 3189 . . 3 𝑦 ∈ V
2 sbcangOLD 38218 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅)))
31, 2e0a 38478 . 2 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
4 sbcel1gvOLD 38574 . . . 4 (𝑦 ∈ V → ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎))
51, 4e0a 38478 . . 3 ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
6 sbceqg 3956 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅))
71, 6e0a 38478 . . . 4 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅)
8 csbingOLD 38534 . . . . . . 7 (𝑦 ∈ V → 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥))
91, 8e0a 38478 . . . . . 6 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥)
10 csbconstg 3527 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑎 = 𝑎)
111, 10e0a 38478 . . . . . . 7 𝑦 / 𝑥𝑎 = 𝑎
12 csbvarg 3975 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
131, 12e0a 38478 . . . . . . 7 𝑦 / 𝑥𝑥 = 𝑦
1411, 13ineq12i 3790 . . . . . 6 (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = (𝑎𝑦)
159, 14eqtri 2643 . . . . 5 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
16 csbconstg 3527 . . . . . 6 (𝑦 ∈ V → 𝑦 / 𝑥∅ = ∅)
171, 16e0a 38478 . . . . 5 𝑦 / 𝑥∅ = ∅
1815, 17eqeq12i 2635 . . . 4 (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅ ↔ (𝑎𝑦) = ∅)
197, 18bitri 264 . . 3 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎𝑦) = ∅)
205, 19anbi12i 732 . 2 (([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
213, 20bitri 264 1 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  [wsbc 3417  csb 3514  cin 3554  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-in 3562
This theorem is referenced by: (None)
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