Theorem opth2 5443

Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)

Hypotheses

Ref	Expression
opth2.1	⊢ 𝐶 ∈ V
opth2.2	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
opth2	⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem opth2

Step	Hyp	Ref	Expression
1		opth2.1	. 2 ⊢ 𝐶 ∈ V
2		opth2.2	. 2 ⊢ 𝐷 ∈ V
3		opthg2 5442	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4	1, 2, 3	mp2an 692	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599

This theorem is referenced by:  eqvinop  5450  opelxp  5677  fsn  7110  opiota  8041  canthwe  10611  ltresr  11100  mat1dimelbas  22365  fmucndlem  24185  hgt750lemb  34654  diblsmopel  41172  cdlemn7  41204  dihordlem7  41215  xihopellsmN  41255  dihopellsm  41256  dihpN  41337  cofidvala  49109  cofidval  49112