Theorem rexlimdv 2650

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 1577	. 2 |- F/ x ph
2		nfv 1577	. 2 |- F/ x ch
3		rexlimdv.1	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
4	1, 2, 3	rexlimd 2648	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   e. wcel 2202   E.wrex 2512

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  rexlimdva  2651  rexlimdv3a  2653  rexlimdva2  2654  rexlimdvw  2655  rexlimdvv  2658  ssorduni  4591  funcnvuni  5406  dffo3  5802  smoiun  6510  tfrlem9  6528  ordiso2  7294  axprecex  8160  recexap  8892  zdiv  9629  btwnz  9660  lbzbi  9911  imasmnd2  13615  imasgrp2  13777  imasrng  14050  imasring  14158  neibl  15302  metcnp3  15322  ushgredgedg  16167  ushgredgedgloop  16169