Theorem abidnf 2894

Description: Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)

Assertion

Ref	Expression
abidnf	|- ( F/_ x A -> { z | A. x z e. A } = A )

Distinct variable groups:   x, z   z, A

Allowed substitution hint:   A( x)

Proof of Theorem abidnf

Step	Hyp	Ref	Expression
1		sp 1499	. . 3 |- ( A. x z e. A -> z e. A )
2		nfcr 2300	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	2	nfrd 1508	. . 3 |- ( F/_ x A -> ( z e. A -> A. x z e. A ) )
4	1, 3	impbid2 142	. 2 |- ( F/_ x A -> ( A. x z e. A <-> z e. A ) )
5	4	abbi1dv 2286	1 |- ( F/_ x A -> { z | A. x z e. A } = A )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   e. wcel 2136   {cab 2151   F/_wnfc 2295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297

This theorem is referenced by:  dedhb  2895  nfopd  3775  nfimad  4955  nffvd  5498