Theorem rexpr 3674

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 3670	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
6	1, 2, 5	mp2an 426	1 |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   {cpr 3619

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-sn 3624  df-pr 3625

This theorem is referenced by: (None)