Theorem abidnf 2898

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 1504	. . 3 |- ( A. x z e. A -> z e. A )
2		nfcr 2304	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	2	nfrd 1513	. . 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 2290	1 |- ( F/_ x A -> { z | A. x z e. A } = A )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   e. wcel 2141   {cab 2156   F/_wnfc 2299

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301

This theorem is referenced by:  dedhb  2899  nfopd  3782  nfimad  4962  nffvd  5508