Theorem r19.23 2642

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 2641	. 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 1509   A.wral 2511   E.wrex 2512

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  r19.23v  2643  rexlimi  2644  rexlimd  2648