Theorem dfrab3ss 3450

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 3178	. . 3 |- ( A C_ B <-> ( A i^i B ) = A )
2		ineq1 3366	. . . 4 |- ( ( A i^i B ) = A -> ( ( A i^i B ) i^i { x | ph } ) = ( A i^i { x | ph } ) )
3	2	eqcomd 2210	. . 3 |- ( ( A i^i B ) = A -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
4	1, 3	sylbi 121	. 2 |- ( A C_ B -> ( A i^i { x | ph } ) = ( ( A i^i B ) i^i { x | ph } ) )
5		dfrab3 3448	. 2 |- { x e. A | ph } = ( A i^i { x | ph } )
6		dfrab3 3448	. . . 4 |- { x e. B | ph } = ( B i^i { x | ph } )
7	6	ineq2i 3370	. . 3 |- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
8		inass 3382	. . 3 |- ( ( A i^i B ) i^i { x | ph } ) = ( A i^i ( B i^i { x | ph } ) )
9	7, 8	eqtr4i 2228	. 2 |- ( A i^i { x e. B | ph } ) = ( ( A i^i B ) i^i { x | ph } )
10	4, 5, 9	3eqtr4g 2262	1 |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Syntax hints:   -> wi 4   = wceq 1372   {cab 2190   {crab 2487   i^i cin 3164   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492  df-v 2773  df-in 3171  df-ss 3178

This theorem is referenced by: (None)