Theorem dvelimdc 2242

Description: Deduction form of dvelimc 2243. (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 1462	. . 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 2236	. . . . 5 |- ( ph -> F/ x w e. A )
6		dvelimdc.4	. . . . . 6 |- ( ph -> F/_ z B )
7	6	nfcrd 2236	. . . . 5 |- ( ph -> F/ z w e. B )
8		dvelimdc.5	. . . . . 6 |- ( ph -> ( z = y -> A = B ) )
9		eleq2 2146	. . . . . 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 1935	. . . 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 2218	. 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 1283   = wceq 1285   F/wnf 1390   e. wcel 1434   F/_wnfc 2210

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212

This theorem is referenced by:  dvelimc  2243