Theorem r19.21v 2396

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 1421	. 2 |- F/ x ph
2	1	r19.21 2395	1 |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by:  r19.32vdc  2459  rmo4  2734  rmo3  2849  dftr5  3857  reusv3  4192  tfrlem1  5923  tfrlemi1  5946  tfri3  5953  rdgon  5973  ordiso2  6357  raluz2  8522