Theorem altopthbg 34197

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 34190	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}))
2		sneqbg 4771	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐶} ↔ 𝐴 = 𝐶))
3		sneqbg 4771	. . . 4 ⊢ (𝐷 ∈ 𝑊 → ({𝐷} = {𝐵} ↔ 𝐷 = 𝐵))
4		eqcom 2745	. . . 4 ⊢ ({𝐵} = {𝐷} ↔ {𝐷} = {𝐵})
5		eqcom 2745	. . . 4 ⊢ (𝐵 = 𝐷 ↔ 𝐷 = 𝐵)
6	3, 4, 5	3bitr4g 313	. . 3 ⊢ (𝐷 ∈ 𝑊 → ({𝐵} = {𝐷} ↔ 𝐵 = 𝐷))
7	2, 6	bi2anan9 635	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (({𝐴} = {𝐶} ∧ {𝐵} = {𝐷}) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
8	1, 7	syl5bb 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {csn 4558  ⟪caltop 34185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561  df-altop 34187

This theorem is referenced by:  altopthb  34199