Theorem opthg 4295

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 3828	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	eqeq1d 2215	. . 3 ⊢ (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩))
3		eqeq1 2213	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ↔ 𝐴 = 𝐶))
4	3	anbi1d 465	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)))
5	2, 4	bibi12d 235	. 2 ⊢ (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷))))
6		opeq2 3829	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
7	6	eqeq1d 2215	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩))
8		eqeq1 2213	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐷 ↔ 𝐵 = 𝐷))
9	8	anbi2d 464	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 = 𝐶 ∧ 𝑦 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
10	7, 9	bibi12d 235	. 2 ⊢ (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝑦 = 𝐷)) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
11		vex 2776	. . 3 ⊢ 𝑥 ∈ V
12		vex 2776	. . 3 ⊢ 𝑦 ∈ V
13	11, 12	opth 4294	. 2 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))
14	5, 10, 13	vtocl2g 2839	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ⟨cop 3641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647

This theorem is referenced by:  opthg2  4296  xpopth  6280  eqop  6281  inl11  7188  preqlu  7615  cauappcvgprlemladd  7801  elrealeu  7972  s111  11118  qnumdenbi  12599  crth  12631  imasaddfnlemg  13231