Theorem r19.21v 2571

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 1539	. 2 |- F/ x ph
2	1	r19.21 2570	1 |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wral 2472

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477

This theorem is referenced by:  r19.32vdc  2643  rmo4  2954  rmo3  3078  dftr5  4131  reusv3  4492  tfrlem1  6363  tfrlemi1  6387  tfr1onlemaccex  6403  tfrcllemaccex  6416  tfri3  6422  ordiso2  7096  raluz2  9647  ndvdssub  12074  nninfalllem1  15568  nninfsellemqall  15575