Theorem altopeq12 35918

Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Assertion

Ref	Expression
altopeq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Proof of Theorem altopeq12

Step	Hyp	Ref	Expression
1		sneq 4658	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2		sneq 4658	. . 3 ⊢ (𝐶 = 𝐷 → {𝐶} = {𝐷})
3	1, 2	anim12i 612	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
4		altopthsn 35917	. 2 ⊢ (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
5	3, 4	sylibr 234	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  {csn 4648  ⟪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:  altopeq1  35919  altopeq2  35920