Theorem altopthc 35600

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 35598 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 2735	. 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 35599	. 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
5		eqcom 2735	. . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6		eqcom 2735	. . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷)
7	5, 6	anbi12i 626	. 2 ⊢ ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
8	1, 4, 7	3bitri 296	1 ⊢ (⟪𝐴, 𝐵⟫ = ⟪𝐶, 𝐷⟫ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473  ⟪caltop 35585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-sn 4633  df-pr 4635  df-altop 35587

This theorem is referenced by: (None)