Theorem dfrab3ss 3242

Description: Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)

Assertion

Ref	Expression
dfrab3ss	|- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem dfrab3ss

Step	Hyp	Ref	Expression
1		df-ss 2986	. . 3 |- ( A C_ B <-> ( A i^i B ) = A )
2		ineq1 3160	. . . 4 |- ( ( A i^i B ) = A -> ( ( A i^i B ) i^i { x | ph } ) = ( A i^i { x | ph } ) )
3	2	eqcomd 2086	. . 3 |- ( ( A i^i B ) = A -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
4	1, 3	sylbi 119	. 2 |- ( A C_ B -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
5		dfrab3 3240	. 2 |- { x e. A | ph } = ( A i^i { x | ph } )
6		dfrab3 3240	. . . 4 |- { x e. B | ph } = ( B i^i { x | ph } )
7	6	ineq2i 3164	. . 3 |- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
8		inass 3176	. . 3 |- ( ( A i^i B ) i^i { x | ph } ) = ( A i^i ( B i^i { x | ph } ) )
9	7, 8	eqtr4i 2104	. 2 |- ( A i^i { x e. B | ph } ) = ( ( A i^i B ) i^i { x | ph } )
10	4, 5, 9	3eqtr4g 2138	1 |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Syntax hints:   -> wi 4   = wceq 1284   {cab 2067   {crab 2352   i^i cin 2972   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986

This theorem is referenced by: (None)