Theorem dvelimdc 2238

Description: Deduction form of dvelimc 2239. (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 1461	. . 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 2232	. . . . 5 |- ( ph -> F/ x w e. A )
6		dvelimdc.4	. . . . . 6 |- ( ph -> F/_ z B )
7	6	nfcrd 2232	. . . . 5 |- ( ph -> F/ z w e. B )
8		dvelimdc.5	. . . . . 6 |- ( ph -> ( z = y -> A = B ) )
9		eleq2 2142	. . . . . 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 1933	. . . 4 |- ( ph -> ( -. A. x x = y -> F/ x w e. B ) )
12	11	imp 122	. . 3 |- ( ( ph /\ -. A. x x = y ) -> F/ x w e. B )
13	1, 12	nfcd 2214	. 2 |- ( ( ph /\ -. A. x x = y ) -> F/_ x B )
14	13	ex 113	1 |- ( ph -> ( -. A. x x = y -> F/_ x B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   F/wnf 1389   e. wcel 1433   F/_wnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by:  dvelimc  2239