Theorem sbcrel 4769

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 3573	. . 3 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) ) )
2		csbconstg 3111	. . . 4 |- ( A e. V -> [_ A / x ]_ ( _V X. _V ) = ( _V X. _V ) )
3	2	sseq2d 3227	. . 3 |- ( A e. V -> ( [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
4	1, 3	bitrd 188	. 2 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ ( _V X. _V ) ) )
5		df-rel 4690	. . 3 |- ( Rel R <-> R C_ ( _V X. _V ) )
6	5	sbcbii 3062	. 2 |- ( [. A / x ]. Rel R <-> [. A / x ]. R C_ ( _V X. _V ) )
7		df-rel 4690	. 2 |- ( Rel [_ A / x ]_ R <-> [_ A / x ]_ R C_ ( _V X. _V ) )
8	4, 6, 7	3bitr4g 223	1 |- ( A e. V -> ( [. A / x ]. Rel R <-> Rel [_ A / x ]_ R ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2177   _Vcvv 2773   [.wsbc 3002   [_csb 3097   C_ wss 3170   X. cxp 4681   Rel wrel 4688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003  df-csb 3098  df-in 3176  df-ss 3183  df-rel 4690

This theorem is referenced by:  sbcfung  5304