Theorem altopthd 36115

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 36112 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Hypotheses

Ref	Expression
altopthd.1	⊢ 𝐶 ∈ V
altopthd.2	⊢ 𝐷 ∈ V

Assertion

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

Proof of Theorem altopthd

Step	Hyp	Ref	Expression
1		eqcom 2741	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫)
2		altopthd.1	. . 3 ⊢ 𝐶 ∈ V
3		altopthd.2	. . 3 ⊢ 𝐷 ∈ V
4	2, 3	altopth 36112	. 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
5		eqcom 2741	. . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6		eqcom 2741	. . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷)
7	5, 6	anbi12i 628	. 2 ⊢ ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
8	1, 4, 7	3bitri 297	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟪caltop 36099

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-sn 4579  df-pr 4581  df-altop 36101

This theorem is referenced by: (None)