Theorem nfceqdf 2760

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 2687	. . . 4 |- ( ph -> ( y e. A <-> y e. B ) )
4	1, 3	nfbidf 2092	. . 3 |- ( ph -> ( F/ x y e. A <-> F/ x y e. B ) )
5	4	albidv 1849	. 2 |- ( ph -> ( A. y F/ x y e. A <-> A. y F/ x y e. B ) )
6		df-nfc 2753	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
7		df-nfc 2753	. 2 |- ( F/_ x B <-> A. y F/ x y e. B )
8	5, 6, 7	3bitr4g 303	1 |- ( ph -> ( F/_ x A <-> F/_ x B ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  nfceqi  2761  nfopd  4419  dfnfc2  4454  dfnfc2OLD  4455  nfimad  5475  nffvd  6200  riotasv2d  34243  nfcxfrdf  34253  nfded  34254  nfded2  34255  nfunidALT2  34256