Theorem r19.23v 2481

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 1466	. 2 |- F/ x ps
2	1	r19.23 2480	1 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wral 2359   E.wrex 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364  df-rex 2365

This theorem is referenced by:  uniiunlem  3109  dfiin2g  3763  iunss  3771  ralxfr2d  4286  rexxfr2d  4287  ssrel2  4528  reliun  4558  funimaexglem  5097  funimass4  5355  ralrnmpt2  5759