Theorem ralim 2685

Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)

Assertion

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

Proof of Theorem ralim

Step	Hyp	Ref	Expression
1		df-ral 2619	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
2		ax-2 7	. . . 4 |- ((x e. A -> (ph -> ps)) -> ((x e. A -> ph) -> (x e. A -> ps)))
3	2	al2imi 1561	. . 3 |- (A.x(x e. A -> (ph -> ps)) -> (A.x(x e. A -> ph) -> A.x(x e. A -> ps)))
4	1, 3	sylbi 187	. 2 |- (A.x e. A (ph -> ps) -> (A.x(x e. A -> ph) -> A.x(x e. A -> ps)))
5		df-ral 2619	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		df-ral 2619	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7	4, 5, 6	3imtr4g 261	1 |- (A.x e. A (ph -> ps) -> (A.x e. A ph -> A.x e. A ps))

Syntax hints:   -> wi 4  A.wal 1540   e. wcel 1710  A.wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ral 2619

This theorem is referenced by:  ral2imi  2690  r19.30  2756