Theorem abidnf 3006

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/_xA -> {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 1747	. . 3 |- (A.x z e. A -> z e. A)
2		nfcr 2482	. . . 4 |- ( F/_xA -> F/x z e. A)
3	2	nfrd 1763	. . 3 |- ( F/_xA -> (z e. A -> A.x z e. A))
4	1, 3	impbid2 195	. 2 |- ( F/_xA -> (A.x z e. A <-> z e. A))
5	4	eqabcdv 2470	1 |- ( F/_xA -> {z | A.x z e. A} = A)

Syntax hints:   -> wi 4  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  dedhb  3007  nfopd  4606  nfimad  4955  nffvd  5336