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Theorem onfrALTlem4 39281
Description: Lemma for onfrALT 39287. (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 3630 . 2 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
2 sbcel1v 3646 . . 3 ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
3 vex 3354 . . . . 5 𝑦 ∈ V
4 sbceqg 4129 . . . . 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 4155 . . . . . 6 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥)
7 csbconstg 3695 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑎 = 𝑎)
83, 7ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑎 = 𝑎
9 csbvarg 4148 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
103, 9ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑥 = 𝑦
118, 10ineq12i 3963 . . . . . 6 (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = (𝑎𝑦)
126, 11eqtri 2793 . . . . 5 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
13 csb0 4127 . . . . 5 𝑦 / 𝑥∅ = ∅
1412, 13eqeq12i 2785 . . . 4 (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅ ↔ (𝑎𝑦) = ∅)
155, 14bitri 264 . . 3 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎𝑦) = ∅)
162, 15anbi12i 612 . 2 (([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
171, 16bitri 264 1 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  [wsbc 3587  csb 3682  cin 3722  c0 4063
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-3an 1073  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-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-in 3730  df-nul 4064
This theorem is referenced by:  onfrALTlem1  39286  onfrALTlem1VD  39646
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