Theorem dfrab3ss 3349

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 3079	. . 3 |- ( A C_ B <-> ( A i^i B ) = A )
2		ineq1 3265	. . . 4 |- ( ( A i^i B ) = A -> ( ( A i^i B ) i^i { x | ph } ) = ( A i^i { x | ph } ) )
3	2	eqcomd 2143	. . 3 |- ( ( A i^i B ) = A -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
4	1, 3	sylbi 120	. 2 |- ( A C_ B -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
5		dfrab3 3347	. 2 |- { x e. A | ph } = ( A i^i { x | ph } )
6		dfrab3 3347	. . . 4 |- { x e. B | ph } = ( B i^i { x | ph } )
7	6	ineq2i 3269	. . 3 |- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
8		inass 3281	. . 3 |- ( ( A i^i B ) i^i { x | ph } ) = ( A i^i ( B i^i { x | ph } ) )
9	7, 8	eqtr4i 2161	. 2 |- ( A i^i { x e. B | ph } ) = ( ( A i^i B ) i^i { x | ph } )
10	4, 5, 9	3eqtr4g 2195	1 |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Syntax hints:   -> wi 4   = wceq 1331   {cab 2123   {crab 2418   i^i cin 3065   C_ wss 3066

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-in 3072  df-ss 3079

This theorem is referenced by: (None)