Theorem tceq 6158

Description: Equality theorem for cardinal T operator. (Contributed by SF, 2-Mar-2015.)

Assertion

Ref	Expression
tceq	|- (A = B -> Tc A = Tc B)

Proof of Theorem tceq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 2808	. . . 4 |- (A = B -> (E.y e. A x = Nc ~P1 y <-> E.y e. B x = Nc ~P1 y))
2	1	anbi2d 684	. . 3 |- (A = B -> ((x e. NC /\ E.y e. A x = Nc ~P1 y) <-> (x e. NC /\ E.y e. B x = Nc ~P1 y)))
3	2	iotabidv 4360	. 2 |- (A = B -> (iotax(x e. NC /\ E.y e. A x = Nc ~P1 y)) = (iotax(x e. NC /\ E.y e. B x = Nc ~P1 y)))
4		df-tc 6103	. 2 |- Tc A = (iotax(x e. NC /\ E.y e. A x = Nc ~P1 y))
5		df-tc 6103	. 2 |- Tc B = (iotax(x e. NC /\ E.y e. B x = Nc ~P1 y))
6	3, 4, 5	3eqtr4g 2410	1 |- (A = B -> Tc A = Tc B)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615  ~P₁ cpw1 4135  iotacio 4337   NC cncs 6088   Nc cnc 6091   T_c ctc 6093

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-uni 3892  df-iota 4339  df-tc 6103

This theorem is referenced by:  tcdi  6164  tc2c  6166  tc11  6228  taddc  6229  tlecg  6230  letc  6231  ce0t  6232  ce2le  6233  cet  6234  tce2  6236  te0c  6237  ce0lenc1  6239  tlenc1c  6240  brtcfn  6246  nmembers1lem1  6268  nmembers1  6271  nchoicelem1  6289  nchoicelem2  6290  nchoicelem12  6300  nchoicelem16  6304  nchoicelem17  6305  nchoicelem19  6307  nchoice  6308