Theorem csbopg 4420

Description: Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.) (Revised by ML, 25-Oct-2020.)

Assertion

Ref	Expression
csbopg	|- ( A e. V -> [_ A / x ]_ <. C , D >. = <. [_ A / x ]_ C , [_ A / x ]_ D >. )

Proof of Theorem csbopg

Step	Hyp	Ref	Expression
1		csbif 4138	. . 3 |- [_ A / x ]_ if ( ( C e. _V /\ D e. _V ) , { { C } , { C , D } } , (/) ) = if ( [. A / x ]. ( C e. _V /\ D e. _V ) , [_ A / x ]_ { { C } , { C , D } } , [_ A / x ]_ (/) )
2		sbcan 3478	. . . . 5 |- ( [. A / x ]. ( C e. _V /\ D e. _V ) <-> ( [. A / x ]. C e. _V /\ [. A / x ]. D e. _V ) )
3		sbcel1g 3987	. . . . . 6 |- ( A e. V -> ( [. A / x ]. C e. _V <-> [_ A / x ]_ C e. _V ) )
4		sbcel1g 3987	. . . . . 6 |- ( A e. V -> ( [. A / x ]. D e. _V <-> [_ A / x ]_ D e. _V ) )
5	3, 4	anbi12d 747	. . . . 5 |- ( A e. V -> ( ( [. A / x ]. C e. _V /\ [. A / x ]. D e. _V ) <-> ( [_ A / x ]_ C e. _V /\ [_ A / x ]_ D e. _V ) ) )
6	2, 5	syl5bb 272	. . . 4 |- ( A e. V -> ( [. A / x ]. ( C e. _V /\ D e. _V ) <-> ( [_ A / x ]_ C e. _V /\ [_ A / x ]_ D e. _V ) ) )
7		csbprg 4244	. . . . 5 |- ( A e. V -> [_ A / x ]_ { { C } , { C , D } } = { [_ A / x ]_ { C } , [_ A / x ]_ { C , D } } )
8		csbsng 4243	. . . . . 6 |- ( A e. V -> [_ A / x ]_ { C } = { [_ A / x ]_ C } )
9		csbprg 4244	. . . . . 6 |- ( A e. V -> [_ A / x ]_ { C , D } = { [_ A / x ]_ C , [_ A / x ]_ D } )
10	8, 9	preq12d 4276	. . . . 5 |- ( A e. V -> { [_ A / x ]_ { C } , [_ A / x ]_ { C , D } } = { { [_ A / x ]_ C } , { [_ A / x ]_ C , [_ A / x ]_ D } } )
11	7, 10	eqtrd 2656	. . . 4 |- ( A e. V -> [_ A / x ]_ { { C } , { C , D } } = { { [_ A / x ]_ C } , { [_ A / x ]_ C , [_ A / x ]_ D } } )
12		csbconstg 3546	. . . 4 |- ( A e. V -> [_ A / x ]_ (/) = (/) )
13	6, 11, 12	ifbieq12d 4113	. . 3 |- ( A e. V -> if ( [. A / x ]. ( C e. _V /\ D e. _V ) , [_ A / x ]_ { { C } , { C , D } } , [_ A / x ]_ (/) ) = if ( ( [_ A / x ]_ C e. _V /\ [_ A / x ]_ D e. _V ) , { { [_ A / x ]_ C } , { [_ A / x ]_ C , [_ A / x ]_ D } } , (/) ) )
14	1, 13	syl5eq 2668	. 2 |- ( A e. V -> [_ A / x ]_ if ( ( C e. _V /\ D e. _V ) , { { C } , { C , D } } , (/) ) = if ( ( [_ A / x ]_ C e. _V /\ [_ A / x ]_ D e. _V ) , { { [_ A / x ]_ C } , { [_ A / x ]_ C , [_ A / x ]_ D } } , (/) ) )
15		dfopif 4399	. . 3 |- <. C , D >. = if ( ( C e. _V /\ D e. _V ) , { { C } , { C , D } } , (/) )
16	15	csbeq2i 3993	. 2 |- [_ A / x ]_ <. C , D >. = [_ A / x ]_ if ( ( C e. _V /\ D e. _V ) , { { C } , { C , D } } , (/) )
17		dfopif 4399	. 2 |- <. [_ A / x ]_ C , [_ A / x ]_ D >. = if ( ( [_ A / x ]_ C e. _V /\ [_ A / x ]_ D e. _V ) , { { [_ A / x ]_ C } , { [_ A / x ]_ C , [_ A / x ]_ D } } , (/) )
18	14, 16, 17	3eqtr4g 2681	1 |- ( A e. V -> [_ A / x ]_ <. C , D >. = <. [_ A / x ]_ C , [_ A / x ]_ D >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   <.cop 4183

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  esum2dlem  30154  csbfinxpg  33225