Theorem p6eq 4238

Description: Equality theorem for P6 operation. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
p6eq	|- (A = B -> P6 A = P6 B)

Proof of Theorem p6eq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3293	. . 3 |- (A = B -> ((_V X.k {{x}}) C_ A <-> (_V X.k {{x}}) C_ B))
2	1	abbidv 2467	. 2 |- (A = B -> {x | (_V X.k {{x}}) C_ A} = {x | (_V X.k {{x}}) C_ B})
3		df-p6 4191	. 2 |- P6 A = {x | (_V X.k {{x}}) C_ A}
4		df-p6 4191	. 2 |- P6 B = {x | (_V X.k {{x}}) C_ B}
5	2, 3, 4	3eqtr4g 2410	1 |- (A = B -> P6 A = P6 B)

Syntax hints:   -> wi 4   = wceq 1642  {cab 2339  _Vcvv 2859   C_ wss 3257  {csn 3737   X._k cxpk 4174   P6 cp6 4178

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-nan 1288  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-p6 4191

This theorem is referenced by:  p6eqi  4239  p6eqd  4240  p6exg  4290