Theorem altopthc 35927

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 35925 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 2747	. 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 35926	. 2 ⊢ (⟪𝐶, 𝐷⟫ = ⟪𝐴, 𝐵⟫ ↔ (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
5		eqcom 2747	. . 3 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6		eqcom 2747	. . 3 ⊢ (𝐷 = 𝐵 ↔ 𝐵 = 𝐷)
7	5, 6	anbi12i 627	. 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 1537   ∈ wcel 2108  Vcvv 3488  ⟪caltop 35912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-sn 4649  df-pr 4651  df-altop 35914

This theorem is referenced by: (None)