New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  nfceqi GIF version

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 (xAxB)

Proof of Theorem nfceqi
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfceqi.1 . . . . 5 A = B
21eleq2i 2417 . . . 4 (y Ay B)
32nfbii 1569 . . 3 (Ⅎx y A ↔ Ⅎx y B)
43albii 1566 . 2 (yx y Ayx y B)
5 df-nfc 2478 . 2 (xAyx y A)
6 df-nfc 2478 . 2 (xByx y B)
74, 5, 63bitr4i 268 1 (xAxB)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
 Copyright terms: Public domain W3C validator