Theorem r19.23 2614

Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)

Hypothesis

Ref	Expression
r19.23.1	|- F/ x ps

Assertion

Ref	Expression
r19.23	|- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Proof of Theorem r19.23

Step	Hyp	Ref	Expression
1		r19.23.1	. 2 |- F/ x ps
2		r19.23t 2613	. 2 |- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )
3	1, 2	ax-mp 5	1 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1483   A.wral 2484   E.wrex 2485

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489  df-rex 2490

This theorem is referenced by:  r19.23v  2615  rexlimi  2616  rexlimd  2620