Theorem dvelimdf 1624

Description: Deduction form of dvelimf 1623. This version may be useful if we want to avoid ax-17 1317 and use ax-16 1580 instead.

Hypotheses

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

Assertion

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

Proof of Theorem dvelimdf

Step	Hyp	Ref	Expression
1		dvelimdf.2	. . . . . 6 |- (ph -> A.zph)
2		dvelimdf.1	. . . . . 6 |- (ph -> A.xph)
3	1, 2	19.21ai 1345	. . . . 5 |- (ph -> A.zA.xph)
4		dvelimdf.3	. . . . . 6 |- (ph -> (ps -> A.xps))
5	4	2alimi 1339	. . . . 5 |- (A.zA.xph -> A.zA.x(ps -> A.xps))
6		hbsb4t 1621	. . . . 5 |- (A.zA.x(ps -> A.xps) -> (-. A.x x = y -> ([y / z]ps -> A.x[y / z]ps)))
7	3, 5, 6	3syl 24	. . . 4 |- (ph -> (-. A.x x = y -> ([y / z]ps -> A.x[y / z]ps)))
8	7	imp 377	. . 3 |- ((ph /\ -. A.x x = y) -> ([y / z]ps -> A.x[y / z]ps))
9		dvelimdf.4	. . . . 5 |- (ph -> (ch -> A.zch))
10		dvelimdf.5	. . . . 5 |- (ph -> (z = y -> (ps <-> ch)))
11	1, 9, 10	sbied 1563	. . . 4 |- (ph -> ([y / z]ps <-> ch))
12	11	adantr 425	. . 3 |- ((ph /\ -. A.x x = y) -> ([y / z]ps <-> ch))
13	2, 11	albid 1459	. . . 4 |- (ph -> (A.x[y / z]ps <-> A.xch))
14	13	adantr 425	. . 3 |- ((ph /\ -. A.x x = y) -> (A.x[y / z]ps <-> A.xch))
15	8, 12, 14	3imtr3d 601	. 2 |- ((ph /\ -. A.x x = y) -> (ch -> A.xch))
16	15	ex 402	1 |- (ph -> (-. A.x x = y -> (ch -> A.xch)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

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