Theorem nfceqdf 2371

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
nfceqdf.1	|- F/ x ph
nfceqdf.2	|- ( ph -> A = B )

Assertion

Ref	Expression
nfceqdf	|- ( ph -> ( F/_ x A <-> F/_ x B ) )

Proof of Theorem nfceqdf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfceqdf.1	. . . 4 |- F/ x ph
2		nfceqdf.2	. . . . 5 |- ( ph -> A = B )
3	2	eleq2d 2299	. . . 4 |- ( ph -> ( y e. A <-> y e. B ) )
4	1, 3	nfbidf 1585	. . 3 |- ( ph -> ( F/ x y e. A <-> F/ x y e. B ) )
5	4	albidv 1870	. 2 |- ( ph -> ( A. y F/ x y e. A <-> A. y F/ x y e. B ) )
6		df-nfc 2361	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
7		df-nfc 2361	. 2 |- ( F/_ x B <-> A. y F/ x y e. B )
8	5, 6, 7	3bitr4g 223	1 |- ( ph -> ( F/_ x A <-> F/_ x B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   e. wcel 2200   F/_wnfc 2359

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-cleq 2222  df-clel 2225  df-nfc 2361

This theorem is referenced by:  nfopd  3874  dfnfc2  3906  nfimad  5077  nffvd  5639