Theorem nceq 6109

Description: Cardinality equality law. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
nceq	|- (A = B -> Nc A = Nc B)

Proof of Theorem nceq

Step	Hyp	Ref	Expression
1		eceq1 5963	. 2 |- (A = B -> [A] ~~ = [B] ~~ )
2		df-nc 6102	. 2 |- Nc A = [A] ~~
3		df-nc 6102	. 2 |- Nc B = [B] ~~
4	1, 2, 3	3eqtr4g 2410	1 |- (A = B -> Nc A = Nc B)

Syntax hints:   -> wi 4   = wceq 1642  [cec 5946   ~~ cen 6029   Nc cnc 6092

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-sn 3742  df-ima 4728  df-ec 5948  df-nc 6102

This theorem is referenced by:  nceqi  6110  nceqd  6111  ncelncs  6121  1cnc  6140  ncaddccl  6145  ce0addcnnul  6180  nc0le1  6217  ce0lenc1  6240  muc0or  6253