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Theorem onfrALTlem4 41742
Description: Lemma for onfrALT 41748. (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 3737 . 2 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ ([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅))
2 sbcel1v 3755 . . 3 ([𝑦 / 𝑥]𝑥𝑎𝑦𝑎)
3 vex 3404 . . . . 5 𝑦 ∈ V
4 sbceqg 4309 . . . . 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 4339 . . . . . 6 𝑦 / 𝑥(𝑎𝑥) = (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥)
7 csbconstg 3819 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑎 = 𝑎)
83, 7ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑎 = 𝑎
9 csbvarg 4331 . . . . . . . 8 (𝑦 ∈ V → 𝑦 / 𝑥𝑥 = 𝑦)
103, 9ax-mp 5 . . . . . . 7 𝑦 / 𝑥𝑥 = 𝑦
118, 10ineq12i 4111 . . . . . 6 (𝑦 / 𝑥𝑎𝑦 / 𝑥𝑥) = (𝑎𝑦)
126, 11eqtri 2762 . . . . 5 𝑦 / 𝑥(𝑎𝑥) = (𝑎𝑦)
13 csb0 4306 . . . . 5 𝑦 / 𝑥∅ = ∅
1412, 13eqeq12i 2754 . . . 4 (𝑦 / 𝑥(𝑎𝑥) = 𝑦 / 𝑥∅ ↔ (𝑎𝑦) = ∅)
155, 14bitri 278 . . 3 ([𝑦 / 𝑥](𝑎𝑥) = ∅ ↔ (𝑎𝑦) = ∅)
162, 15anbi12i 630 . 2 (([𝑦 / 𝑥]𝑥𝑎[𝑦 / 𝑥](𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
171, 16bitri 278 1 ([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wcel 2114  Vcvv 3400  [wsbc 3685  csb 3800  cin 3852  c0 4221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-in 3860  df-nul 4222
This theorem is referenced by:  onfrALTlem1  41747  onfrALTlem1VD  42089
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