Theorem nfbii 1521

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 1518	. . . 4 |- ( A. x ph <-> A. x ps )
3	1, 2	imbi12i 239	. . 3 |- ( ( ph -> A. x ph ) <-> ( ps -> A. x ps ) )
4	3	albii 1518	. 2 |- ( A. x ( ph -> A. x ph ) <-> A. x ( ps -> A. x ps ) )
5		df-nf 1509	. 2 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
6		df-nf 1509	. 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 1395   F/wnf 1508

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 1495  ax-gen 1497

This theorem depends on definitions:  df-bi 117  df-nf 1509

This theorem is referenced by:  nfxfr  1522  nfxfrd  1523  nfsb  1999  nfsbt  2029  hbsbd  2035  sbal1yz  2054  dvelimALT  2063  dvelimfv  2064  dvelimor  2071  nfeudv  2094  nfeuv  2097  nfceqi  2370  nfreudxy  2707  dfnfc2  3911