Theorem dvelimdf 2044

Description: Deduction form of dvelimf 2043. This version may be useful if we want to avoid ax-17 1549 and use ax-16 1837 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Hypotheses

Ref	Expression
dvelimdf.1	|- F/ x ph
dvelimdf.2	|- F/ z ph
dvelimdf.3	|- ( ph -> F/ x ps )
dvelimdf.4	|- ( ph -> F/ z ch )
dvelimdf.5	|- ( ph -> ( z = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
dvelimdf	|- ( ph -> ( -. A. x x = y -> F/ x ch ) )

Proof of Theorem dvelimdf

Step	Hyp	Ref	Expression
1		dvelimdf.2	. . . 4 |- F/ z ph
2		dvelimdf.3	. . . 4 |- ( ph -> F/ x ps )
3	1, 2	alrimi 1545	. . 3 |- ( ph -> A. z F/ x ps )
4		nfsb4t 2042	. . 3 |- ( A. z F/ x ps -> ( -. A. x x = y -> F/ x [ y / z ] ps ) )
5	3, 4	syl 14	. 2 |- ( ph -> ( -. A. x x = y -> F/ x [ y / z ] ps ) )
6		dvelimdf.1	. . 3 |- F/ x ph
7		dvelimdf.4	. . . 4 |- ( ph -> F/ z ch )
8		dvelimdf.5	. . . 4 |- ( ph -> ( z = y -> ( ps <-> ch ) ) )
9	1, 7, 8	sbied 1811	. . 3 |- ( ph -> ( [ y / z ] ps <-> ch ) )
10	6, 9	nfbidf 1562	. 2 |- ( ph -> ( F/ x [ y / z ] ps <-> F/ x ch ) )
11	5, 10	sylibd 149	1 |- ( ph -> ( -. A. x x = y -> F/ x ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1371   F/wnf 1483   [wsb 1785

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786

This theorem is referenced by:  dvelimdc  2369