Theorem rexlimdv 2659

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 2657	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   e. wcel 2203   E.wrex 2521

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 2525  df-rex 2526

This theorem is referenced by:  rexlimdva  2660  rexlimdv3a  2662  rexlimdva2  2663  rexlimdvw  2664  rexlimdvv  2667  ssorduni  4609  funcnvuni  5425  dffo3  5824  smoiun  6532  tfrlem9  6550  ordiso2  7326  axprecex  8195  recexap  8927  zdiv  9666  btwnz  9697  lbzbi  9948  imasmnd2  13665  imasgrp2  13827  imasrng  14100  imasring  14208  neibl  15356  metcnp3  15376  ushgredgedg  16221  ushgredgedgloop  16223