Theorem nceqd 6111

Description: Equality deduction for cardinality. (Contributed by SF, 24-Feb-2015.)

Hypothesis

Ref	Expression
nceqd.1	⊢ (φ → A = B)

Assertion

Ref	Expression
nceqd	⊢ (φ → Nc A = Nc B)

Proof of Theorem nceqd

Step	Hyp	Ref	Expression
1		nceqd.1	. 2 ⊢ (φ → A = B)
2		nceq 6109	. 2 ⊢ (A = B → Nc A = Nc B)
3	1, 2	syl 15	1 ⊢ (φ → Nc A = Nc B)

Syntax hints:   → wi 4   = wceq 1642   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:  df1c3g  6142  ncspw1eu  6160  pw1eltc  6163  dflec2  6211  nchoicelem7  6296  nchoicelem9  6298  nchoicelem11  6300  nchoicelem12  6301  nchoicelem14  6303  nchoicelem15  6304  nchoicelem16  6305  nchoicelem17  6306  nchoice  6309