Theorem altopthbg 34940

Description: Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)

Assertion

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

Proof of Theorem altopthbg

Step	Hyp	Ref	Expression
1		altopthsn 34933	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqbg 4845	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3		sneqbg 4845	. . . 4 ⊢ (𝐷 ∈ 𝑊 → ({𝐷} = {𝐵} ↔ 𝐷 = 𝐵))
4		eqcom 2740	. . . 4 ⊢ ({𝐵} = {𝐷} ↔ {𝐷} = {𝐵})
5		eqcom 2740	. . . 4 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵)
6	3, 4, 5	3bitr4g 314	. . 3 ⊢ (𝐷 ∈ 𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
7	2, 6	bi2anan9 638	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
8	1, 7	bitrid 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {csn 4629  ⟪caltop 34928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-altop 34930

This theorem is referenced by:  altopthb  34942