Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onfrALTlem4 Structured version   Visualization version   GIF version

Theorem onfrALTlem4 40437
 Description: Lemma for onfrALT 40443. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem4 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
Distinct variable group:   𝑥,𝑎

Proof of Theorem onfrALTlem4
StepHypRef Expression
1 sbcan 3755 . 2 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
2 sbcel1v 3773 . . 3 ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
3 vex 3443 . . . . 5 𝑦 ∈ V
4 sbceqg 4287 . . . . 5 (𝑦 ∈ V → ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅))
53, 4ax-mp 5 . . . 4 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ 𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅)
6 csbin 4312 . . . . . 6 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥)
7 csbconstg 3834 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑎 = 𝑎)
83, 7ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑎 = 𝑎
9 csbvarg 4304 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
103, 9ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑥 = 𝑦
118, 10ineq12i 4113 . . . . . 6 (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = (𝑎𝑦)
126, 11eqtri 2821 . . . . 5 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
13 csb0 4285 . . . . 5 𝑦 / 𝑥∅ = ∅
1412, 13eqeq12i 2811 . . . 4 (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅ ↔ (𝑎𝑦) = ∅)
155, 14bitri 276 . . 3 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎𝑦) = ∅)
162, 15anbi12i 626 . 2 (([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
171, 16bitri 276 1 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  Vcvv 3440  [wsbc 3711  ⦋csb 3817   ∩ cin 3864  ∅c0 4217 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-in 3872  df-nul 4218 This theorem is referenced by:  onfrALTlem1  40442  onfrALTlem1VD  40784
 Copyright terms: Public domain W3C validator