Theorem nfceqdf 2177

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 2107	. . . 4 |- ( ph -> ( y e. A <-> y e. B ) )
4	1, 3	nfbidf 1432	. . 3 |- ( ph -> ( F/ x y e. A <-> F/ x y e. B ) )
5	4	albidv 1705	. 2 |- ( ph -> ( A. y F/ x y e. A <-> A. y F/ x y e. B ) )
6		df-nfc 2167	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
7		df-nfc 2167	. 2 |- ( F/_ x B <-> A. y F/ x y e. B )
8	5, 6, 7	3bitr4g 212	1 |- ( ph -> ( F/_ x A <-> F/_ x B ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   F/wnf 1349   e. wcel 1393   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167

This theorem is referenced by:  nfopd  3566  dfnfc2  3598  nfimad  4677  nffvd  5187