Theorem dvelimdc 2395

Description: Deduction form of dvelimc 2396. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
dvelimdc.1	|- F/ x ph
dvelimdc.2	|- F/ z ph
dvelimdc.3	|- ( ph -> F/_ x A )
dvelimdc.4	|- ( ph -> F/_ z B )
dvelimdc.5	|- ( ph -> ( z = y -> A = B ) )

Assertion

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

Proof of Theorem dvelimdc

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1576	. . 3 |- F/ w ( ph /\ -. A. x x = y )
2		dvelimdc.1	. . . . 5 |- F/ x ph
3		dvelimdc.2	. . . . 5 |- F/ z ph
4		dvelimdc.3	. . . . . 6 |- ( ph -> F/_ x A )
5	4	nfcrd 2388	. . . . 5 |- ( ph -> F/ x w e. A )
6		dvelimdc.4	. . . . . 6 |- ( ph -> F/_ z B )
7	6	nfcrd 2388	. . . . 5 |- ( ph -> F/ z w e. B )
8		dvelimdc.5	. . . . . 6 |- ( ph -> ( z = y -> A = B ) )
9		eleq2 2295	. . . . . 6 |- ( A = B -> ( w e. A <-> w e. B ) )
10	8, 9	syl6 33	. . . . 5 |- ( ph -> ( z = y -> ( w e. A <-> w e. B ) ) )
11	2, 3, 5, 7, 10	dvelimdf 2069	. . . 4 |- ( ph -> ( -. A. x x = y -> F/ x w e. B ) )
12	11	imp 124	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/ x w e. B )
13	1, 12	nfcd 2369	. 2 |- ( ( ph /\ -. A. x x = y ) -> F/_ x B )
14	13	ex 115	1 |- ( ph -> ( -. A. x x = y -> F/_ x B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1395   = wceq 1397   F/wnf 1508   e. wcel 2202   F/_wnfc 2361

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363

This theorem is referenced by:  dvelimc  2396