Theorem r19.23 2730

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/xps

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/xps
2		r19.23t 2729	. 2 |- ( F/xps -> (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 176   F/wnf 1544  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  r19.23v  2731  rexlimi  2732  rexlimd  2736