Theorem r19.23v 2603

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 1539	. 2 |- F/ x ps
2	1	r19.23 2602	1 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wral 2472   E.wrex 2473

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  uniiunlem  3268  dfiin2g  3945  iunss  3953  ralxfr2d  4495  rexxfr2d  4496  ssrel2  4749  reliun  4780  funimaexglem  5337  funimass4  5607  ralrnmpo  6033