Theorem abidnf 2709

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 1401	. . 3 |- ( A. x z e. A -> z e. A )
2		nfcr 2170	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	2	nfrd 1413	. . 3 |- ( F/_ x A -> ( z e. A -> A. x z e. A ) )
4	1, 3	impbid2 131	. 2 |- ( F/_ x A -> ( A. x z e. A <-> z e. A ) )
5	4	abbi1dv 2157	1 |- ( F/_ x A -> { z | A. x z e. A } = A )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167

This theorem is referenced by:  dedhb  2710  nfopd  3566  nfimad  4677  nffvd  5187