Theorem abidnf 3375

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 2053	. . 3 |- ( A. x z e. A -> z e. A )
2		nfcr 2756	. . . 4 |- ( F/_ x A -> F/ x z e. A )
3	2	nf5rd 2066	. . 3 |- ( F/_ x A -> ( z e. A -> A. x z e. A ) )
4	1, 3	impbid2 216	. 2 |- ( F/_ x A -> ( A. x z e. A <-> z e. A ) )
5	4	abbi1dv 2743	1 |- ( F/_ x A -> { z | A. x z e. A } = A )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  dedhb  3376  nfopd  4419  nfimad  5475  nffvd  6200  nfunidALT2  34256  nfunidALT  34257  nfopdALT  34258