Theorem altopthc 36396

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 36394 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Hypotheses

Ref	Expression
altopthc.1	⊢ 𝐵 ∈ V
altopthc.2	⊢ 𝐶 ∈ V

Assertion

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

Proof of Theorem altopthc

Step	Hyp	Ref	Expression
1		eqcom 2776	. 2 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ ⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫)
2		altopthc.2	. . 3 ⊢ 𝐶 ∈ V
3		altopthc.1	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	altopthb 36395	. 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
5		eqcom 2776	. . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6		eqcom 2776	. . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷)
7	5, 6	anbi12i 639	. 2 ⊢ ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
8	1, 4, 7	3bitri 300	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟪caltop 36381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-sn 4595  df-pr 4597  df-altop 36383

This theorem is referenced by: (None)