Theorem r19.21v 2627

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 1581	. 2 |- F/ x ph
2	1	r19.21 2626	1 |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is used by:  r19.32vdc  2700  rmo4  3019  rmo3  3144  dftr5  4232  reusv3  4606  tfrlem1  6579  tfrlemi1  6603  tfr1onlemaccex  6619  tfrcllemaccex  6632  tfri3  6638  ordiso2  7375  raluz2  9979  ndvdssub  12697  nninfalllem1  17051  nninfsellemqall  17058