Theorem nfbii 1407

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 1404	. . . 4 |- ( A. x ph <-> A. x ps )
3	1, 2	imbi12i 237	. . 3 |- ( ( ph -> A. x ph ) <-> ( ps -> A. x ps ) )
4	3	albii 1404	. 2 |- ( A. x ( ph -> A. x ph ) <-> A. x ( ps -> A. x ps ) )
5		df-nf 1395	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
6		df-nf 1395	. 2 |- ( F/ x ps <-> A. x ( ps -> A. x ps ) )
7	4, 5, 6	3bitr4i 210	1 |- ( F/ x ph <-> F/ x ps )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1287   F/wnf 1394

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383

This theorem depends on definitions:  df-bi 115  df-nf 1395

This theorem is referenced by:  nfxfr  1408  nfxfrd  1409  nfsb  1870  nfsbt  1898  hbsbd  1906  sbal1yz  1925  dvelimALT  1934  dvelimfv  1935  dvelimor  1942  nfeudv  1963  nfeuv  1966  nfceqi  2224  nfreudxy  2540  dfnfc2  3671