Theorem bnj1534 28930

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 2540	. . 3 |- F/_ x A
3		nfcv 2540	. . 3 |- F/_ z A
4		nfv 1626	. . 3 |- F/ z ( F ` x ) =/= ( H ` x )
5		bnj1534.2	. . . . . 6 |- ( w e. F -> A. x w e. F )
6	5	nfcii 2531	. . . . 5 |- F/_ x F
7		nfcv 2540	. . . . 5 |- F/_ x z
8	6, 7	nffv 5694	. . . 4 |- F/_ x ( F ` z )
9		nfcv 2540	. . . 4 |- F/_ x ( H ` z )
10	8, 9	nfne 2658	. . 3 |- F/ x ( F ` z ) =/= ( H ` z )
11		fveq2 5687	. . . 4 |- ( x = z -> ( F ` x ) = ( F ` z ) )
12		fveq2 5687	. . . 4 |- ( x = z -> ( H ` x ) = ( H ` z ) )
13	11, 12	neeq12d 2582	. . 3 |- ( x = z -> ( ( F ` x ) =/= ( H ` x ) <-> ( F ` z ) =/= ( H ` z ) ) )
14	2, 3, 4, 10, 13	cbvrab 2914	. 2 |- { x e. A | ( F ` x ) =/= ( H ` x ) } = { z e. A | ( F ` z ) =/= ( H ` z ) }
15	1, 14	eqtri 2424	1 |- D = { z e. A | ( F ` z ) =/= ( H ` z ) }

Syntax hints:   -> wi 4   A.wal 1546   = wceq 1649   e. wcel 1721   =/= wne 2567   {crab 2670   ` cfv 5413

This theorem is referenced by:  bnj1523  29146

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421