Theorem nfbii 1471

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 1468	. . . 4 |- ( A. x ph <-> A. x ps )
3	1, 2	imbi12i 239	. . 3 |- ( ( ph -> A. x ph ) <-> ( ps -> A. x ps ) )
4	3	albii 1468	. 2 |- ( A. x ( ph -> A. x ph ) <-> A. x ( ps -> A. x ps ) )
5		df-nf 1459	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
6		df-nf 1459	. 2 |- ( F/ x ps <-> A. x ( ps -> A. x ps ) )
7	4, 5, 6	3bitr4i 212	1 |- ( F/ x ph <-> F/ x ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   F/wnf 1458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447

This theorem depends on definitions:  df-bi 117  df-nf 1459

This theorem is referenced by:  nfxfr  1472  nfxfrd  1473  nfsb  1944  nfsbt  1974  hbsbd  1980  sbal1yz  1999  dvelimALT  2008  dvelimfv  2009  dvelimor  2016  nfeudv  2039  nfeuv  2042  nfceqi  2313  nfreudxy  2648  dfnfc2  3823