Theorem opeqex 4251

Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)

Assertion

Ref	Expression
opeqex	|- ( <. A , B >. = <. C , D >. -> ( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) )

Proof of Theorem opeqex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2241	. . 3 |- ( <. A , B >. = <. C , D >. -> ( x e. <. A , B >. <-> x e. <. C , D >. ) )
2	1	exbidv 1825	. 2 |- ( <. A , B >. = <. C , D >. -> ( E. x x e. <. A , B >. <-> E. x x e. <. C , D >. ) )
3		opm 4236	. 2 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4		opm 4236	. 2 |- ( E. x x e. <. C , D >. <-> ( C e. _V /\ D e. _V ) )
5	2, 3, 4	3bitr3g 222	1 |- ( <. A , B >. = <. C , D >. -> ( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2739   <.cop 3597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603

This theorem is referenced by:  epelg  4292