Theorem rexlimdv 2624

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 1552	. 2 |- F/ x ph
2		nfv 1552	. 2 |- F/ x ch
3		rexlimdv.1	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
4	1, 2, 3	rexlimd 2622	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   e. wcel 2178   E.wrex 2487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2491  df-rex 2492

This theorem is referenced by:  rexlimdva  2625  rexlimdv3a  2627  rexlimdva2  2628  rexlimdvw  2629  rexlimdvv  2632  ssorduni  4553  funcnvuni  5362  dffo3  5750  smoiun  6410  tfrlem9  6428  ordiso2  7163  axprecex  8028  recexap  8761  zdiv  9496  btwnz  9527  lbzbi  9772  imasmnd2  13399  imasgrp2  13561  imasrng  13833  imasring  13941  neibl  15078  metcnp3  15098