Theorem altopthg 36392

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)

Assertion

Ref	Expression
altopthg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem altopthg

Step	Hyp	Ref	Expression
1		altopthsn 36386	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqbg 4812	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3		sneqbg 4812	. . 3 ⊢ (𝐵 ∈ 𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
4	2, 3	bi2anan9 649	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
5	1, 4	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {csn 4594  ⟪caltop 36381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-sn 4595  df-pr 4597  df-altop 36383

This theorem is referenced by:  altopth  36394