Theorem rexlimdv 2546

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Hypothesis

Ref	Expression
rexlimdv.1	|- ( ph -> ( x e. A -> ( ps -> ch ) ) )

Assertion

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

Distinct variable groups:   ph, x   ch, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem rexlimdv

Step	Hyp	Ref	Expression
1		nfv 1508	. 2 |- F/ x ph
2		nfv 1508	. 2 |- F/ x ch
3		rexlimdv.1	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
4	1, 2, 3	rexlimd 2544	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   e. wcel 1480   E.wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420

This theorem is referenced by:  rexlimdva  2547  rexlimdv3a  2549  rexlimdva2  2550  rexlimdvw  2551  rexlimdvv  2554  ssorduni  4398  funcnvuni  5187  dffo3  5560  smoiun  6191  tfrlem9  6209  ordiso2  6913  axprecex  7681  recexap  8407  zdiv  9132  btwnz  9163  lbzbi  9401  neibl  12649  metcnp3  12669