Theorem altopthbg 33429

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 33422	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqbg 4774	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3		sneqbg 4774	. . . 4 ⊢ (𝐷 ∈ 𝑊 → ({𝐷} = {𝐵} ↔ 𝐷 = 𝐵))
4		eqcom 2828	. . . 4 ⊢ ({𝐵} = {𝐷} ↔ {𝐷} = {𝐵})
5		eqcom 2828	. . . 4 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵)
6	3, 4, 5	3bitr4g 316	. . 3 ⊢ (𝐷 ∈ 𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
7	2, 6	bi2anan9 637	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
8	1, 7	syl5bb 285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {csn 4567  ⟪caltop 33417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4568  df-pr 4570  df-altop 33419

This theorem is referenced by:  altopthb  33431