Theorem bnj1534 29298

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 2574	. . 3 |- F/_ x A
3		nfcv 2574	. . 3 |- F/_ z A
4		nfv 1630	. . 3 |- F/ z ( F ` x ) =/= ( H ` x )
5		bnj1534.2	. . . . . 6 |- ( w e. F -> A. x w e. F )
6	5	nfcii 2565	. . . . 5 |- F/_ x F
7		nfcv 2574	. . . . 5 |- F/_ x z
8	6, 7	nffv 5738	. . . 4 |- F/_ x ( F ` z )
9		nfcv 2574	. . . 4 |- F/_ x ( H ` z )
10	8, 9	nfne 2697	. . 3 |- F/ x ( F ` z ) =/= ( H ` z )
11		fveq2 5731	. . . 4 |- ( x = z -> ( F ` x ) = ( F ` z ) )
12		fveq2 5731	. . . 4 |- ( x = z -> ( H ` x ) = ( H ` z ) )
13	11, 12	neeq12d 2618	. . 3 |- ( x = z -> ( ( F ` x ) =/= ( H ` x ) <-> ( F ` z ) =/= ( H ` z ) ) )
14	2, 3, 4, 10, 13	cbvrab 2956	. 2 |- { x e. A | ( F ` x ) =/= ( H ` x ) } = { z e. A | ( F ` z ) =/= ( H ` z ) }
15	1, 14	eqtri 2458	1 |- D = { z e. A | ( F ` z ) =/= ( H ` z ) }

Syntax hints:   -> wi 4   A.wal 1550   = wceq 1653   e. wcel 1726   =/= wne 2601   {crab 2711   ` cfv 5457

This theorem is referenced by:  bnj1523  29514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465