Theorem rexpr 3448

Description: Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralpr.1	|- A e. _V
ralpr.2	|- B e. _V
ralpr.3	|- ( x = A -> ( ph <-> ps ) )
ralpr.4	|- ( x = B -> ( ph <-> ch ) )

Assertion

Ref	Expression
rexpr	|- ( E. x e. { A , B } ph <-> ( ps \/ ch ) )

Distinct variable groups:   x, A   x, B   ps, x   ch, x

Allowed substitution hint:   ph( x)

Proof of Theorem rexpr

Step	Hyp	Ref	Expression
1		ralpr.1	. 2 |- A e. _V
2		ralpr.2	. 2 |- B e. _V
3		ralpr.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
4		ralpr.4	. . 3 |- ( x = B -> ( ph <-> ch ) )
5	3, 4	rexprg 3444	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
6	1, 2, 5	mp2an 416	1 |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) )

Syntax hints:   -> wi 4   <-> wb 103   \/ wo 661   = wceq 1284   e. wcel 1433   E.wrex 2349   _Vcvv 2601   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405

This theorem is referenced by: (None)