Theorem difopabOLD 5821

Description: Obsolete version of difopab 5820 as of 19-Dec-2024. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
difopabOLD	⊢ ({⟨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 difopabOLD

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabv 5811	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		reldif 5805	. . 3 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
3	1, 2	ax-mp 5	. 2 ⊢ Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4		relopabv 5811	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
5		sbcan 3820	. . . 4 ⊢ ([𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
6		sbcan 3820	. . . . 5 ⊢ ([𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ ([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓))
7	6	sbcbii 3827	. . . 4 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓) ↔ [𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦] ¬ 𝜓))
8		opelopabsb 5515	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9		sbcng 3818	. . . . . . 7 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓))
10	9	elv 3468	. . . . . 6 ⊢ ([𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓 ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
11		sbcng 3818	. . . . . . . 8 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓))
12	11	elv 3468	. . . . . . 7 ⊢ ([𝑤 / 𝑦] ¬ 𝜓 ↔ ¬ [𝑤 / 𝑦]𝜓)
13	12	sbcbii 3827	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ [𝑤 / 𝑦]𝜓)
14		opelopabsb 5515	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
15	14	notbii 320	. . . . . 6 ⊢ (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ¬ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
16	10, 13, 15	3bitr4ri 304	. . . . 5 ⊢ (¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓)
17	8, 16	anbi12i 628	. . . 4 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦] ¬ 𝜓))
18	5, 7, 17	3bitr4ri 304	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
19		eldif 3941	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ¬ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
20		opelopabsb 5515	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)} ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ ¬ 𝜓))
21	18, 19, 20	3bitr4i 303	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)})
22	3, 4, 21	eqrelriiv 5780	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463  [wsbc 3770   ∖ cdif 3928  ⟨cop 4612  {copab 5185  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5186  df-xp 5671  df-rel 5672

This theorem is referenced by: (None)