Theorem nceq 6108

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 5962	. 2 ⊢ (A = B → [A] ≈ = [B] ≈ )
2		df-nc 6101	. 2 ⊢ Nc A = [A] ≈
3		df-nc 6101	. 2 ⊢ Nc B = [B] ≈
4	1, 2, 3	3eqtr4g 2410	1 ⊢ (A = B → Nc A = Nc B)

Syntax hints:   → wi 4   = wceq 1642  [cec 5945   ≈ cen 6028   Nc cnc 6091

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 2478  df-rex 2620  df-sn 3741  df-ima 4727  df-ec 5947  df-nc 6101

This theorem is referenced by:  nceqi  6109  nceqd  6110  ncelncs  6120  1cnc  6139  ncaddccl  6144  ce0addcnnul  6179  nc0le1  6216  ce0lenc1  6239  muc0or  6252