Theorem rexpr 3626

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 3622	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
6	1, 2, 5	mp2an 423	1 |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698   = wceq 1342   e. wcel 2135   E.wrex 2443   _Vcvv 2721   {cpr 3571

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-sn 3576  df-pr 3577

This theorem is referenced by: (None)