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Theorem difopab 5175
 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 5169 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 reldif 5161 . . 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 5169 . 2 Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
5 sbcan 3445 . . . 4 ([𝑧 / 𝑥]([𝑤 / 𝑦]𝜑[𝑤 / 𝑦] ¬ 𝜓) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
6 sbcan 3445 . . . . 5 ([𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ ([𝑤 / 𝑦]𝜑[𝑤 / 𝑦] ¬ 𝜓))
76sbcbii 3458 . . . 4 ([𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ [𝑧 / 𝑥]([𝑤 / 𝑦]𝜑[𝑤 / 𝑦] ¬ 𝜓))
8 opelopabsb 4910 . . . . 5 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9 vex 3176 . . . . . . 7 𝑧 ∈ V
10 sbcng 3443 . . . . . . 7 (𝑧 ∈ V → ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓))
119, 10ax-mp 5 . . . . . 6 ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
12 vex 3176 . . . . . . . 8 𝑤 ∈ V
13 sbcng 3443 . . . . . . . 8 (𝑤 ∈ V → ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓))
1412, 13ax-mp 5 . . . . . . 7 ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓)
1514sbcbii 3458 . . . . . 6 ([𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓[𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓)
16 opelopabsb 4910 . . . . . . 7 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
1716notbii 309 . . . . . 6 (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
1811, 15, 173bitr4ri 292 . . . . 5 (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓)
198, 18anbi12i 729 . . . 4 ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
205, 7, 193bitr4ri 292 . . 3 ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
21 eldif 3550 . . 3 (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
22 opelopabsb 4910 . . 3 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
2320, 21, 223bitr4i 291 . 2 (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)})
243, 4, 23eqrelriiv 5137 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402   ∖ cdif 3537  ⟨cop 4131  {copab 4642  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by: (None)
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