Theorem r19.20 1251

Description: Distribution of restricted quantification over implication.

Assertion

Ref	Expression
r19.20	|- (A.x e. A (ph -> ps) -> (A.x e. A ph -> A.x e. A ps))

Proof of Theorem r19.20

Step	Hyp	Ref	Expression
1		df-ral 1205	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
2		ax-2 4	. . . 4 |- ((x e. A -> (ph -> ps)) -> ((x e. A -> ph) -> (x e. A -> ps)))
3	2	19.20ii 692	. . 3 |- (A.x(x e. A -> (ph -> ps)) -> (A.x(x e. A -> ph) -> A.x(x e. A -> ps)))
4	1, 3	sylbi 174	. 2 |- (A.x e. A (ph -> ps) -> (A.x(x e. A -> ph) -> A.x(x e. A -> ps)))
5		df-ral 1205	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		df-ral 1205	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7	4, 5, 6	3imtr4g 426	1 |- (A.x e. A (ph -> ps) -> (A.x e. A ph -> A.x e. A ps))

Syntax hints:   -> wi 2  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  tfrlem1 2949  tz7.49 2997  abianfp 3000  bnd 3548  kmlem11 3590  osumlem4 5533

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

metamath.org