Theorem r19.23v 2586

Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Assertion

Ref	Expression
r19.23v	|- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Distinct variable group:   ps, x

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

Proof of Theorem r19.23v

Step	Hyp	Ref	Expression
1		nfv 1528	. 2 |- F/ x ps
2	1	r19.23 2585	1 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wral 2455   E.wrex 2456

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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  uniiunlem  3245  dfiin2g  3920  iunss  3928  ralxfr2d  4465  rexxfr2d  4466  ssrel2  4717  reliun  4748  funimaexglem  5300  funimass4  5567  ralrnmpo  5989