Theorem abidnf 2821

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 1471	. . 3 |- ( A. x z e. A -> z e. A )
2		nfcr 2247	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	2	nfrd 1483	. . 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 2234	1 |- ( F/_ x A -> { z | A. x z e. A } = A )

Syntax hints:   -> wi 4   A.wal 1312   = wceq 1314   e. wcel 1463   {cab 2101   F/_wnfc 2242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244

This theorem is referenced by:  dedhb  2822  nfopd  3688  nfimad  4848  nffvd  5387