Theorem r19.21v 2509

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.21v	|- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Distinct variable group:   ph, x

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

Proof of Theorem r19.21v

Step	Hyp	Ref	Expression
1		nfv 1508	. 2 |- F/ x ph
2	1	r19.21 2508	1 |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wral 2416

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 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421

This theorem is referenced by:  r19.32vdc  2580  rmo4  2877  rmo3  3000  dftr5  4029  reusv3  4381  tfrlem1  6205  tfrlemi1  6229  tfr1onlemaccex  6245  tfrcllemaccex  6258  tfri3  6264  ordiso2  6920  raluz2  9374  ndvdssub  11627  nninfalllem1  13203  nninfsellemqall  13211