Theorem nfceqi 2485

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/_xA <-> F/_xB)

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 2417	. . . 4 |- (y e. A <-> y e. B)
3	2	nfbii 1569	. . 3 |- ( F/x y e. A <-> F/x y e. B)
4	3	albii 1566	. 2 |- (A.y F/x y e. A <-> A.y F/x y e. B)
5		df-nfc 2478	. 2 |- ( F/_xA <-> A.y F/x y e. A)
6		df-nfc 2478	. 2 |- ( F/_xB <-> A.y F/x y e. B)
7	4, 5, 6	3bitr4i 268	1 |- ( F/_xA <-> F/_xB)

Syntax hints:   <-> wb 176  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  nfcxfr  2486  nfcxfrd  2487