Theorem dvelimdf 2335

Description: Deduction form of dvelimf 2334. (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.1	. . . 4 |- F/ x ph
2		dvelimdf.3	. . . 4 |- ( ph -> F/ x ps )
3	1, 2	nfim1 2067	. . 3 |- F/ x ( ph -> ps )
4		dvelimdf.2	. . . 4 |- F/ z ph
5		dvelimdf.4	. . . 4 |- ( ph -> F/ z ch )
6	4, 5	nfim1 2067	. . 3 |- F/ z ( ph -> ch )
7		dvelimdf.5	. . . . 5 |- ( ph -> ( z = y -> ( ps <-> ch ) ) )
8	7	com12 32	. . . 4 |- ( z = y -> ( ph -> ( ps <-> ch ) ) )
9	8	pm5.74d 262	. . 3 |- ( z = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
10	3, 6, 9	dvelimf 2334	. 2 |- ( -. A. x x = y -> F/ x ( ph -> ch ) )
11		pm5.5 351	. . 3 |- ( ph -> ( ( ph -> ch ) <-> ch ) )
12	1, 11	nfbidf 2092	. 2 |- ( ph -> ( F/ x ( ph -> ch ) <-> F/ x ch ) )
13	10, 12	syl5ib 234	1 |- ( ph -> ( -. A. x x = y -> F/ x ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  nfsb4t  2389  dvelimdc  2786