Theorem altopthd 32079

Description: Alternate ordered pair theorem with different sethood requirements. See altopth 32076 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Hypotheses

Ref	Expression
altopthd.1	|- C e. _V
altopthd.2	|- D e. _V

Assertion

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

Proof of Theorem altopthd

Step	Hyp	Ref	Expression
1		eqcom 2629	. 2 |- ( << A , B >> = << C , D >> <-> << C , D >> = << A , B >> )
2		altopthd.1	. . 3 |- C e. _V
3		altopthd.2	. . 3 |- D e. _V
4	2, 3	altopth 32076	. 2 |- ( << C , D >> = << A , B >> <-> ( C = A /\ D = B ) )
5		eqcom 2629	. . 3 |- ( C = A <-> A = C )
6		eqcom 2629	. . 3 |- ( D = B <-> B = D )
7	5, 6	anbi12i 733	. 2 |- ( ( C = A /\ D = B ) <-> ( A = C /\ B = D ) )
8	1, 4, 7	3bitri 286	1 |- ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <<caltop 32063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-altop 32065

This theorem is referenced by: (None)