Theorem nfceqdf 2391

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 2308	. . . 4 |- ( ph -> ( y e. A <-> y e. B ) )
4	1, 3	nfbidf 1592	. . 3 |- ( ph -> ( F/ x y e. A <-> F/ x y e. B ) )
5	4	albidv 1877	. 2 |- ( ph -> ( A. y F/ x y e. A <-> A. y F/ x y e. B ) )
6		df-nfc 2381	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
7		df-nfc 2381	. 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 1400   = wceq 1402   F/wnf 1513   e. wcel 2209   F/_wnfc 2379

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-clel 2234  df-nfc 2381

This theorem is referenced by:  nfopd  3916  dfnfc2  3948  nfimad  5130  nffvd  5702