Theorem r19.21t 2700

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)

Assertion

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

Proof of Theorem r19.21t

Step	Hyp	Ref	Expression
1		bi2.04 350	. . . 4 |- ((x e. A -> (ph -> ps)) <-> (ph -> (x e. A -> ps)))
2	1	albii 1566	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x(ph -> (x e. A -> ps)))
3		19.21t 1795	. . 3 |- ( F/xph -> (A.x(ph -> (x e. A -> ps)) <-> (ph -> A.x(x e. A -> ps))))
4	2, 3	syl5bb 248	. 2 |- ( F/xph -> (A.x(x e. A -> (ph -> ps)) <-> (ph -> A.x(x e. A -> ps))))
5		df-ral 2620	. 2 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
6		df-ral 2620	. . 3 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7	6	imbi2i 303	. 2 |- ((ph -> A.x e. A ps) <-> (ph -> A.x(x e. A -> ps)))
8	4, 5, 7	3bitr4g 279	1 |- ( F/xph -> (A.x e. A (ph -> ps) <-> (ph -> A.x e. A ps)))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   e. wcel 1710  A.wral 2615

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-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  r19.21  2701