Theorem opthg2 4374

Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opthg2	⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem opthg2

Step	Hyp	Ref	Expression
1		opthg 4373	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
2		eqcom 2240	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
3		eqcom 2240	. . 3 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
4		eqcom 2240	. . 3 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵)
5	3, 4	anbi12i 464	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
6	1, 2, 5	3bitr4g 223	1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ⟨cop 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  opth2  4375  fliftel  5989  axprecex  8237