Theorem nfceqdf 2254

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 2184	. . . 4 |- ( ph -> ( y e. A <-> y e. B ) )
4	1, 3	nfbidf 1502	. . 3 |- ( ph -> ( F/ x y e. A <-> F/ x y e. B ) )
5	4	albidv 1778	. 2 |- ( ph -> ( A. y F/ x y e. A <-> A. y F/ x y e. B ) )
6		df-nfc 2244	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
7		df-nfc 2244	. 2 |- ( F/_ x B <-> A. y F/ x y e. B )
8	5, 6, 7	3bitr4g 222	1 |- ( ph -> ( F/_ x A <-> F/_ x B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1312   = wceq 1314   F/wnf 1419   e. wcel 1463   F/_wnfc 2242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-cleq 2108  df-clel 2111  df-nfc 2244

This theorem is referenced by:  nfopd  3688  dfnfc2  3720  nfimad  4848  nffvd  5387