Theorem nfceqi 2215

Description: Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfceqi.1	|- A = B

Assertion

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

Proof of Theorem nfceqi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfceqi.1	. . . . 5 |- A = B
2	1	eleq2i 2145	. . . 4 |- ( y e. A <-> y e. B )
3	2	nfbii 1402	. . 3 |- ( F/ x y e. A <-> F/ x y e. B )
4	3	albii 1399	. 2 |- ( A. y F/ x y e. A <-> A. y F/ x y e. B )
5		df-nfc 2208	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
6		df-nfc 2208	. 2 |- ( F/_ x B <-> A. y F/ x y e. B )
7	4, 5, 6	3bitr4i 210	1 |- ( F/_ x A <-> F/_ x B )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   F/wnf 1389   e. wcel 1433   F/_wnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by:  nfcxfr  2216  nfcxfrd  2217