Theorem altopthb 35965

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 35964 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Hypotheses

Ref	Expression
altopthb.1	⊢ 𝐴 ∈ V
altopthb.2	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
altopthb	⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Proof of Theorem altopthb

Step	Hyp	Ref	Expression
1		altopthb.1	. 2 ⊢ 𝐴 ∈ V
2		altopthb.2	. 2 ⊢ 𝐷 ∈ V
3		altopthbg 35963	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐷 ∈ V) → (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
4	1, 2, 3	mp2an 692	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟪caltop 35951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-sn 4593  df-pr 4595  df-altop 35953

This theorem is referenced by:  altopthc  35966