Theorem r19.21 2546

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)

Hypothesis

Ref	Expression
r19.21.1	|- F/ x ph

Assertion

Ref	Expression
r19.21	|- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Proof of Theorem r19.21

Step	Hyp	Ref	Expression
1		r19.21.1	. 2 |- F/ x ph
2		r19.21t 2545	. 2 |- ( F/ x ph -> ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) )
3	1, 2	ax-mp 5	1 |- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1453   A.wral 2448

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 1440  ax-gen 1442  ax-4 1503  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453

This theorem is referenced by:  r19.21v  2547  rmo3f  2927