Theorem sbcrel 4730

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 3547	. . 3 |- ( A e. V -> ( [. A / x ]. R C_ ( _V X. _V ) <-> [_ A / x ]_ R C_ [_ A / x ]_ ( _V X. _V ) ) )
2		csbconstg 3086	. . . 4 |- ( A e. V -> [_ A / x ]_ ( _V X. _V ) = ( _V X. _V ) )
3	2	sseq2d 3200	. . 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 4651	. . 3 |- ( Rel R <-> R C_ ( _V X. _V ) )
6	5	sbcbii 3037	. 2 |- ( [. A / x ]. Rel R <-> [. A / x ]. R C_ ( _V X. _V ) )
7		df-rel 4651	. 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 2160   _Vcvv 2752   [.wsbc 2977   [_csb 3072   C_ wss 3144   X. cxp 4642   Rel wrel 4649

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073  df-in 3150  df-ss 3157  df-rel 4651

This theorem is referenced by:  sbcfung  5259