Theorem dvelimdc 2510

Description: Deduction form of dvelimc 2511. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
dvelimdc.1	|- F/xph
dvelimdc.2	|- F/zph
dvelimdc.3	|- (ph -> F/_xA)
dvelimdc.4	|- (ph -> F/_zB)
dvelimdc.5	|- (ph -> (z = y -> A = B))

Assertion

Ref	Expression
dvelimdc	|- (ph -> (-. A.x x = y -> F/_xB))

Proof of Theorem dvelimdc

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 |- F/w(ph /\ -. A.x x = y)
2		dvelimdc.1	. . . . 5 |- F/xph
3		dvelimdc.2	. . . . 5 |- F/zph
4		dvelimdc.3	. . . . . 6 |- (ph -> F/_xA)
5	4	nfcrd 2503	. . . . 5 |- (ph -> F/x w e. A)
6		dvelimdc.4	. . . . . 6 |- (ph -> F/_zB)
7	6	nfcrd 2503	. . . . 5 |- (ph -> F/z w e. B)
8		dvelimdc.5	. . . . . 6 |- (ph -> (z = y -> A = B))
9		eleq2 2414	. . . . . 6 |- (A = B -> (w e. A <-> w e. B))
10	8, 9	syl6 29	. . . . 5 |- (ph -> (z = y -> (w e. A <-> w e. B)))
11	2, 3, 5, 7, 10	dvelimdf 2082	. . . 4 |- (ph -> (-. A.x x = y -> F/x w e. B))
12	11	imp 418	. . 3 |- ((ph /\ -. A.x x = y) -> F/x w e. B)
13	1, 12	nfcd 2485	. 2 |- ((ph /\ -. A.x x = y) -> F/_xB)
14	13	ex 423	1 |- (ph -> (-. A.x x = y -> F/_xB))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  dvelimc  2511