Theorem sbcrel 4690

Description: Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Assertion

Ref	Expression
sbcrel	|- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) )

Proof of Theorem sbcrel

Step	Hyp	Ref	Expression
1		sbcssg 3518	. . 3 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) ) )
2		csbconstg 3059	. . . 4 |- ( A e. V -> [_ A / x ]_ ( _V X. _V ) = ( _V X. _V ) )
3	2	sseq2d 3172	. . 3 |- ( A e. V -> ( [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
4	1, 3	bitrd 187	. 2 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
5		df-rel 4611	. . 3 |- ( Rel R <-> R C_ ( _V X. _V ) )
6	5	sbcbii 3010	. 2 |- ( [. A / x ]. Rel R <-> [. A / x ]. R C_ ( _V X. _V ) )
7		df-rel 4611	. 2 |- ( Rel [_ A / x ]_ R <-> [_ A / x ]_ R C_ ( _V X. _V ) )
8	4, 6, 7	3bitr4g 222	1 |- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   _Vcvv 2726   [.wsbc 2951   [_csb 3045   C_ wss 3116   X. cxp 4602   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046  df-in 3122  df-ss 3129  df-rel 4611

This theorem is referenced by:  sbcfung  5212