Theorem bnj1534 30923

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1534.1	|- D = { x e. A | ( F ` x ) =/= ( H ` x ) }
bnj1534.2	|- ( w e. F -> A. x w e. F )

Assertion

Ref	Expression
bnj1534	|- D = { z e. A | ( F ` z ) =/= ( H ` z ) }

Distinct variable groups:   w, A, x, z   w, F, z   w, H, x, z

Allowed substitution hints:   D( x, z, w)   F( x)

Proof of Theorem bnj1534

Step	Hyp	Ref	Expression
1		bnj1534.1	. 2 |- D = { x e. A | ( F ` x ) =/= ( H ` x ) }
2		nfcv 2764	. . 3 |- F/_ x A
3		nfcv 2764	. . 3 |- F/_ z A
4		nfv 1843	. . 3 |- F/ z ( F ` x ) =/= ( H ` x )
5		bnj1534.2	. . . . . 6 |- ( w e. F -> A. x w e. F )
6	5	nfcii 2755	. . . . 5 |- F/_ x F
7		nfcv 2764	. . . . 5 |- F/_ x z
8	6, 7	nffv 6198	. . . 4 |- F/_ x ( F ` z )
9		nfcv 2764	. . . 4 |- F/_ x ( H ` z )
10	8, 9	nfne 2894	. . 3 |- F/ x ( F ` z ) =/= ( H ` z )
11		fveq2 6191	. . . 4 |- ( x = z -> ( F ` x ) = ( F ` z ) )
12		fveq2 6191	. . . 4 |- ( x = z -> ( H ` x ) = ( H ` z ) )
13	11, 12	neeq12d 2855	. . 3 |- ( x = z -> ( ( F ` x ) =/= ( H ` x ) <-> ( F ` z ) =/= ( H ` z ) ) )
14	2, 3, 4, 10, 13	cbvrab 3198	. 2 |- { x e. A | ( F ` x ) =/= ( H ` x ) } = { z e. A | ( F ` z ) =/= ( H ` z ) }
15	1, 14	eqtri 2644	1 |- D = { z e. A | ( F ` z ) =/= ( H ` z ) }

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   ` cfv 5888

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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj1523  31139