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Theorem onfrALTlem4 39278
Description: Lemma for onfrALT 39284. (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 3619 . 2 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
2 sbcel1v 3636 . . 3 ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
3 vex 3343 . . . . 5 𝑦 ∈ V
4 sbceqg 4127 . . . . 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 4153 . . . . . 6 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥)
7 csbconstg 3687 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑎 = 𝑎)
83, 7ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑎 = 𝑎
9 csbvarg 4146 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
103, 9ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑥 = 𝑦
118, 10ineq12i 3955 . . . . . 6 (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = (𝑎𝑦)
126, 11eqtri 2782 . . . . 5 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
13 csb0 4125 . . . . 5 𝑦 / 𝑥∅ = ∅
1412, 13eqeq12i 2774 . . . 4 (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅ ↔ (𝑎𝑦) = ∅)
155, 14bitri 264 . . 3 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎𝑦) = ∅)
162, 15anbi12i 735 . 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 383   = wceq 1632  wcel 2139  Vcvv 3340  [wsbc 3576  csb 3674  cin 3714  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-in 3722  df-nul 4059
This theorem is referenced by:  onfrALTlem1  39283  onfrALTlem1VD  39643
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