Theorem rexsng 3674

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)

Hypothesis

Ref	Expression
ralsng.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rexsng	|- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem rexsng

Step	Hyp	Ref	Expression
1		rexsns 3672	. 2 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3031	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	bitrid 192	1 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   E.wrex 2485   [.wsbc 2998   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999  df-sn 3639

This theorem is referenced by:  rexsn  3677  rexprg  3685  rextpg  3687  iunxsng  4003  imasng  5047  dvdsprmpweqnn  12659  mnd1  13287  grp1  13438