Theorem opeqex 4282

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 2260	. . 3 |- ( <. A , B >. = <. C , D >. -> ( x e. <. A , B >. <-> x e. <. C , D >. ) )
2	1	exbidv 1839	. 2 |- ( <. A , B >. = <. C , D >. -> ( E. x x e. <. A , B >. <-> E. x x e. <. C , D >. ) )
3		opm 4267	. 2 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4		opm 4267	. 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 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   <.cop 3625

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631

This theorem is referenced by:  epelg  4325