Theorem ralimdaa 2170

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90.

Hypotheses

Ref	Expression
ralimdaa.1	|- (ph -> A.xph)
ralimdaa.2	|- ((ph /\ x e. A) -> (ps -> ch))

Assertion

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

Proof of Theorem ralimdaa

Step	Hyp	Ref	Expression
1		ralimdaa.1	. . 3 |- (ph -> A.xph)
2		ralimdaa.2	. . . . 5 |- ((ph /\ x e. A) -> (ps -> ch))
3	2	ex 402	. . . 4 |- (ph -> (x e. A -> (ps -> ch)))
4	3	a2d 16	. . 3 |- (ph -> ((x e. A -> ps) -> (x e. A -> ch)))
5	1, 4	alimd 1343	. 2 |- (ph -> (A.x(x e. A -> ps) -> A.x(x e. A -> ch)))
6		df-ral 2109	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7		df-ral 2109	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
8	5, 6, 7	3imtr4g 612	1 |- (ph -> (A.x e. A ps -> A.x e. A ch))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   e. wcel 1300  A.wral 2105

This theorem is referenced by:  ralimdvaa 2171  uniiunlem 2693  fopab2 4796  clm4lei 8341  fopab2g 14485  npincppr 14501  fprodneg 14741  taralt 15211  mettrifi 15847

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321

This theorem depends on definitions:  df-bi 164  df-an 242  df-ral 2109

Copyright terms: Public domain