Theorem eloprabga 7525

Description: The law of concretion for operation class abstraction. Compare elopab 5523. (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) Avoid ax-10 2129, ax-11 2146. (Revised by Wolf Lammen, 15-Oct-2024.)

Hypothesis

Ref	Expression
eloprabga.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
eloprabga	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem eloprabga

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3482	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3482	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		elex 3482	. 2 ⊢ (𝐶 ∈ 𝑋 → 𝐶 ∈ V)
4		opex 5460	. . 3 ⊢ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ V
5		simpr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
6	5	eqeq1d 2727	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
7		eqcom 2732	. . . . . . . . . 10 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
8		vex 3467	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
9		vex 3467	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
10		vex 3467	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
11	8, 9, 10	otth2 5479	. . . . . . . . . 10 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶))
12	7, 11	bitri 274	. . . . . . . . 9 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶))
13	6, 12	bitrdi 286	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶)))
14	13	anbi1d 629	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑)))
15		eloprabga.1	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
16	15	pm5.32i 573	. . . . . . 7 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓))
17	14, 16	bitrdi 286	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓)))
18	17	3exbidv 1920	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓)))
19		df-oprab 7420	. . . . . . . . 9 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
20	19	eleq2i 2817	. . . . . . . 8 ⊢ (𝑤 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝑤 ∈ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)})
21		abid 2706	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
22	20, 21	bitr2i 275	. . . . . . 7 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ 𝑤 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
23		eleq1 2813	. . . . . . 7 ⊢ (𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (𝑤 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}))
24	22, 23	bitrid 282	. . . . . 6 ⊢ (𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}))
25	24	adantl 480	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}))
26		19.41vvv 1947	. . . . . . 7 ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓))
27		elisset 2807	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → ∃𝑥 𝑥 = 𝐴)
28		elisset 2807	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → ∃𝑦 𝑦 = 𝐵)
29		elisset 2807	. . . . . . . . . 10 ⊢ (𝐶 ∈ V → ∃𝑧 𝑧 = 𝐶)
30	27, 28, 29	3anim123i 1148	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
31		3exdistr 1956	. . . . . . . . . . 11 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)))
32		19.41v 1945	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)))
33		19.41v 1945	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) ↔ (∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
34	33	anbi2i 621	. . . . . . . . . . 11 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)) ↔ (∃𝑥 𝑥 = 𝐴 ∧ (∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)))
35	31, 32, 34	3bitri 296	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ (∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)))
36		3anass 1092	. . . . . . . . . 10 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ (∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶)))
37	35, 36	bitr4i 277	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
38	30, 37	sylibr 233	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶))
39	38	biantrurd 531	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝜓 ↔ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓)))
40	26, 39	bitr4id 289	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓) ↔ 𝜓))
41	40	adantr 479	. . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜓) ↔ 𝜓))
42	18, 25, 41	3bitr3d 308	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
43	42	expcom 412	. . 3 ⊢ (𝑤 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓)))
44	4, 43	vtocle 3533	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
45	1, 2, 3, 44	syl3an 1157	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  Vcvv 3463  ⟨cop 4630  {coprab 7417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-oprab 7420

This theorem is referenced by:  eloprabg  7527  ovigg  7563  vdwpc  16948  elmpps  35240  uncov  37131  brrabga  37869