Theorem altopthd 35953

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 35950 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 35950	. 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 1536   ∈ wcel 2105  Vcvv 3477  ⟪caltop 35937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-altop 35939

This theorem is referenced by: (None)