Theorem nfceqi 2174

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 2104	. . . 4 |- ( y e. A <-> y e. B )
3	2	nfbii 1362	. . 3 |- ( F/ x y e. A <-> F/ x y e. B )
4	3	albii 1359	. 2 |- ( A. y F/ x y e. A <-> A. y F/ x y e. B )
5		df-nfc 2167	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
6		df-nfc 2167	. 2 |- ( F/_ x B <-> A. y F/ x y e. B )
7	4, 5, 6	3bitr4i 201	1 |- ( F/_ x A <-> F/_ x B )

Syntax hints:   <-> 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:  nfcxfr  2175  nfcxfrd  2176