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Theorem altopeq12 32044
Description: Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopeq12 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)

Proof of Theorem altopeq12
StepHypRef Expression
1 sneq 4178 . . 3 (𝐴 = 𝐵 → {𝐴} = {𝐵})
2 sneq 4178 . . 3 (𝐶 = 𝐷 → {𝐶} = {𝐷})
31, 2anim12i 589 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
4 altopthsn 32043 . 2 (⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫ ↔ ({𝐴} = {𝐵} ∧ {𝐶} = {𝐷}))
53, 4sylibr 224 1 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟪𝐴, 𝐶⟫ = ⟪𝐵, 𝐷⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  {csn 4168  caltop 32038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-pr 4171  df-altop 32040
This theorem is referenced by:  altopeq1  32045  altopeq2  32046
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