Theorem nfceqi 2388

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 2305	. . . 4 |- ( y e. A <-> y e. B )
3	2	nfbii 1526	. . 3 |- ( F/ x y e. A <-> F/ x y e. B )
4	3	albii 1523	. 2 |- ( A. y F/ x y e. A <-> A. y F/ x y e. B )
5		df-nfc 2381	. 2 |- ( F/_ x A <-> A. y F/ x y e. A )
6		df-nfc 2381	. 2 |- ( F/_ x B <-> A. y F/ x y e. B )
7	4, 5, 6	3bitr4i 212	1 |- ( F/_ x A <-> F/_ x B )

Syntax hints:   <-> 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:  nfcxfr  2389  nfcxfrd  2390