Theorem opeqex 4227

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 2230	. . 3 |- ( <. A , B >. = <. C , D >. -> ( x e. <. A , B >. <-> x e. <. C , D >. ) )
2	1	exbidv 1813	. 2 |- ( <. A , B >. = <. C , D >. -> ( E. x x e. <. A , B >. <-> E. x x e. <. C , D >. ) )
3		opm 4212	. 2 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4		opm 4212	. 2 |- ( E. x x e. <. C , D >. <-> ( C e. _V /\ D e. _V ) )
5	2, 3, 4	3bitr3g 221	1 |- ( <. A , B >. = <. C , D >. -> ( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   <.cop 3579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585

This theorem is referenced by:  epelg  4268