Theorem r19.21v 2534

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 2533	1 |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wral 2435

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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440

This theorem is referenced by:  r19.32vdc  2606  rmo4  2905  rmo3  3028  dftr5  4065  reusv3  4420  tfrlem1  6255  tfrlemi1  6279  tfr1onlemaccex  6295  tfrcllemaccex  6308  tfri3  6314  ordiso2  6979  raluz2  9490  ndvdssub  11821  nninfalllem1  13591  nninfsellemqall  13598