Theorem nfbii 1432

Description: Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfbii.1	|- ( ph <-> ps )

Assertion

Ref	Expression
nfbii	|- ( F/ x ph <-> F/ x ps )

Proof of Theorem nfbii

Step	Hyp	Ref	Expression
1		nfbii.1	. . . 4 |- ( ph <-> ps )
2	1	albii 1429	. . . 4 |- ( A. x ph <-> A. x ps )
3	1, 2	imbi12i 238	. . 3 |- ( ( ph -> A. x ph ) <-> ( ps -> A. x ps ) )
4	3	albii 1429	. 2 |- ( A. x ( ph -> A. x ph ) <-> A. x ( ps -> A. x ps ) )
5		df-nf 1420	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
6		df-nf 1420	. 2 |- ( F/ x ps <-> A. x ( ps -> A. x ps ) )
7	4, 5, 6	3bitr4i 211	1 |- ( F/ x ph <-> F/ x ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1312   F/wnf 1419

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-gen 1408

This theorem depends on definitions:  df-bi 116  df-nf 1420

This theorem is referenced by:  nfxfr  1433  nfxfrd  1434  nfsb  1897  nfsbt  1925  hbsbd  1933  sbal1yz  1952  dvelimALT  1961  dvelimfv  1962  dvelimor  1969  nfeudv  1990  nfeuv  1993  nfceqi  2251  nfreudxy  2578  dfnfc2  3720