Theorem dvelimc 2787

Description: Version of dvelim 2337 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem dvelimc

Step	Hyp	Ref	Expression
1		nftru 1730	. . 3 |- F/ x T.
2		nftru 1730	. . 3 |- F/ z T.
3		dvelimc.1	. . . 4 |- F/_ x A
4	3	a1i 11	. . 3 |- ( T. -> F/_ x A )
5		dvelimc.2	. . . 4 |- F/_ z B
6	5	a1i 11	. . 3 |- ( T. -> F/_ z B )
7		dvelimc.3	. . . 4 |- ( z = y -> A = B )
8	7	a1i 11	. . 3 |- ( T. -> ( z = y -> A = B ) )
9	1, 2, 4, 6, 8	dvelimdc 2786	. 2 |- ( T. -> ( -. A. x x = y -> F/_ x B ) )
10	9	trud 1493	1 |- ( -. A. x x = y -> F/_ x B )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   = wceq 1483   T. wtru 1484   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  nfcvf  2788