Theorem opthg 5386

Description: Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opthg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem opthg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4801	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	eqeq1d 2740	. . 3 ⊢ (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩))
3		eqeq1 2742	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶))
4	3	anbi1d 629	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)))
5	2, 4	bibi12d 345	. 2 ⊢ (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷))))
6		opeq2 4802	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
7	6	eqeq1d 2740	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩))
8		eqeq1 2742	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷))
9	8	anbi2d 628	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
10	7, 9	bibi12d 345	. 2 ⊢ (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
11		vex 3426	. . 3 ⊢ 𝑥 ∈ V
12		vex 3426	. . 3 ⊢ 𝑦 ∈ V
13	11, 12	opth 5385	. 2 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))
14	5, 10, 13	vtocl2g 3500	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565

This theorem is referenced by:  opth1g  5387  opthg2  5388  opthneg  5390  otthg  5394  oteqex  5408  s111  14248  embedsetcestrclem  17790  symg2bas  18915  frgpnabllem1  19389  frgpnabllem2  19390  mat1dimbas  21529  linds2eq  31477  goeleq12bg  33211  opideq  36405  dvheveccl  39053  hoidmv1le  44022  oppr  44411  opprb  44412  fsetsnf1  44433  prproropf1olem4  44846