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Theorem difopab 5695
Description: The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
difopab ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem difopab
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5689 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 reldif 5681 . . 3 (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
31, 2ax-mp 5 . 2 Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4 relopab 5689 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
5 sbcan 3818 . . . 4 ([𝑧 / 𝑥]([𝑤 / 𝑦]𝜑[𝑤 / 𝑦] ¬ 𝜓) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
6 sbcan 3818 . . . . 5 ([𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ ([𝑤 / 𝑦]𝜑[𝑤 / 𝑦] ¬ 𝜓))
76sbcbii 3826 . . . 4 ([𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ [𝑧 / 𝑥]([𝑤 / 𝑦]𝜑[𝑤 / 𝑦] ¬ 𝜓))
8 opelopabsb 5408 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9 sbcng 3816 . . . . . . 7 (𝑧 ∈ V → ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓))
109elv 3497 . . . . . 6 ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
11 sbcng 3816 . . . . . . . 8 (𝑤 ∈ V → ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓))
1211elv 3497 . . . . . . 7 ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓)
1312sbcbii 3826 . . . . . 6 ([𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓[𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓)
14 opelopabsb 5408 . . . . . . 7 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
1514notbii 321 . . . . . 6 (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
1610, 13, 153bitr4ri 305 . . . . 5 (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓)
178, 16anbi12i 626 . . . 4 ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
185, 7, 173bitr4ri 305 . . 3 ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
19 eldif 3943 . . 3 (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
20 opelopabsb 5408 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
2118, 19, 203bitr4i 304 . 2 (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)})
223, 4, 21eqrelriiv 5656 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  [wsbc 3769  cdif 3930  cop 4563  {copab 5119  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  dfnelbr2  43349
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