Theorem r19.23v 2615

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 1551	. 2 |- F/ x ps
2	1	r19.23 2614	1 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wral 2484   E.wrex 2485

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489  df-rex 2490

This theorem is referenced by:  uniiunlem  3282  dfiin2g  3960  iunss  3968  ralxfr2d  4511  rexxfr2d  4512  ssrel2  4765  reliun  4796  funimaexglem  5357  funimass4  5629  ralrnmpo  6060