Theorem altopeq12 34264

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 4571	. . 3 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
2		sneq 4571	. . 3 ⊢ (𝐶 = 𝐷 → {𝐶} = {𝐷})
3	1, 2	anim12i 613	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
4		altopthsn 34263	. 2 ⊢ (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
5	3, 4	sylibr 233	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  {csn 4561  ⟪caltop 34258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-altop 34260

This theorem is referenced by:  altopeq1  34265  altopeq2  34266