Theorem r19.21v 2543

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 1516	. 2 |- F/ x ph
2	1	r19.21 2542	1 |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wral 2444

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449

This theorem is referenced by:  r19.32vdc  2615  rmo4  2919  rmo3  3042  dftr5  4083  reusv3  4438  tfrlem1  6276  tfrlemi1  6300  tfr1onlemaccex  6316  tfrcllemaccex  6329  tfri3  6335  ordiso2  7000  raluz2  9517  ndvdssub  11867  nninfalllem1  13888  nninfsellemqall  13895