Theorem altopth 28167

Description: The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that C and D are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4677), requires D to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)

Hypotheses

Ref	Expression
altopth.1	|- A e. _V
altopth.2	|- B e. _V

Assertion

Ref	Expression
altopth	|- ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) )

Proof of Theorem altopth

Step	Hyp	Ref	Expression
1		altopth.1	. 2 |- A e. _V
2		altopth.2	. 2 |- B e. _V
3		altopthg 28165	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) ) )
4	1, 2, 3	mp2an 672	1 |- ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3078   <<caltop 28154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-sn 3989  df-pr 3991  df-altop 28156

This theorem is referenced by:  altopthd  28170  altopelaltxp  28174