Theorem ralim 2380

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 2311	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		ax-2 6	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) -> ( ( x e. A -> ph ) -> ( x e. A -> ps ) ) )
3	2	al2imi 1347	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) -> ( A. x ( x e. A -> ph ) -> A. x ( x e. A -> ps ) ) )
4	1, 3	sylbi 114	. 2 |- ( A. x e. A ( ph -> ps ) -> ( A. x ( x e. A -> ph ) -> A. x ( x e. A -> ps ) ) )
5		df-ral 2311	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
6		df-ral 2311	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7	4, 5, 6	3imtr4g 194	1 |- ( A. x e. A ( ph -> ps ) -> ( A. x e. A ph -> A. x e. A ps ) )

Syntax hints:   -> wi 4   A.wal 1241   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338

This theorem depends on definitions:  df-bi 110  df-ral 2311

This theorem is referenced by:  ral2imi  2385  trint  3869  peano2  4318  mpteqb  5261  lbzbi  8551  r19.29uz  9590