Theorem r19.23v 2575

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 1516	. 2 |- F/ x ps
2	1	r19.23 2574	1 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wral 2444   E.wrex 2445

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-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  uniiunlem  3231  dfiin2g  3899  iunss  3907  ralxfr2d  4442  rexxfr2d  4443  ssrel2  4694  reliun  4725  funimaexglem  5271  funimass4  5537  ralrnmpo  5956