Theorem r19.23 2585

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 2584	. 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 1460   A.wral 2455   E.wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  r19.23v  2586  rexlimi  2587  rexlimd  2591