Theorem dvelimdf 2082

Description: Deduction form of dvelimf 1997. This version may be useful if we want to avoid ax-17 1616 and use ax-16 2144 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem dvelimdf

Step	Hyp	Ref	Expression
1		dvelimdf.2	. . . . . 6 |- F/zph
2		dvelimdf.3	. . . . . 6 |- (ph -> F/xps)
3	1, 2	alrimi 1765	. . . . 5 |- (ph -> A.z F/xps)
4		nfsb4t 2080	. . . . 5 |- (A.z F/xps -> (-. A.x x = y -> F/x[y / z]ps))
5	3, 4	syl 15	. . . 4 |- (ph -> (-. A.x x = y -> F/x[y / z]ps))
6	5	imp 418	. . 3 |- ((ph /\ -. A.x x = y) -> F/x[y / z]ps)
7		dvelimdf.1	. . . . 5 |- F/xph
8		nfnae 1956	. . . . 5 |- F/x -. A.x x = y
9	7, 8	nfan 1824	. . . 4 |- F/x(ph /\ -. A.x x = y)
10		dvelimdf.4	. . . . . 6 |- (ph -> F/zch)
11		dvelimdf.5	. . . . . 6 |- (ph -> (z = y -> (ps <-> ch)))
12	1, 10, 11	sbied 2036	. . . . 5 |- (ph -> ([y / z]ps <-> ch))
13	12	adantr 451	. . . 4 |- ((ph /\ -. A.x x = y) -> ([y / z]ps <-> ch))
14	9, 13	nfbidf 1774	. . 3 |- ((ph /\ -. A.x x = y) -> ( F/x[y / z]ps <-> F/xch))
15	6, 14	mpbid 201	. 2 |- ((ph /\ -. A.x x = y) -> F/xch)
16	15	ex 423	1 |- (ph -> (-. A.x x = y -> F/xch))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  dvelimdc  2510