Theorem brtxpsd2 32002

Description: Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)

Hypotheses

Ref	Expression
brtxpsd2.1	|- A e. _V
brtxpsd2.2	|- B e. _V
brtxpsd2.3	|- R = ( C \ ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) )
brtxpsd2.4	|- A C B

Assertion

Ref	Expression
brtxpsd2	|- ( A R B <-> A. x ( x e. B <-> x S A ) )

Distinct variable groups:   x, A   x, B   x, S

Allowed substitution hints:   C( x)   R( x)

Proof of Theorem brtxpsd2

Step	Hyp	Ref	Expression
1		brtxpsd2.4	. . 3 |- A C B
2		brtxpsd2.3	. . . . 5 |- R = ( C \ ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) )
3	2	breqi 4659	. . . 4 |- ( A R B <-> A ( C \ ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) ) B )
4		brdif 4705	. . . 4 |- ( A ( C \ ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) ) B <-> ( A C B /\ -. A ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) B ) )
5	3, 4	bitri 264	. . 3 |- ( A R B <-> ( A C B /\ -. A ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) B ) )
6	1, 5	mpbiran 953	. 2 |- ( A R B <-> -. A ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) B )
7		brtxpsd2.1	. . 3 |- A e. _V
8		brtxpsd2.2	. . 3 |- B e. _V
9	7, 8	brtxpsd 32001	. 2 |- ( -. A ran ( ( _V (x) _E ) /_\ ( S (x) _V ) ) B <-> A. x ( x e. B <-> x S A ) )
10	6, 9	bitri 264	1 |- ( A R B <-> A. x ( x e. B <-> x S A ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   /_\ csymdif 3843   class class class wbr 4653   _E cep 5028   ran crn 5115   (x) ctxp 31937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961

This theorem is referenced by:  brtxpsd3  32003