Theorem dfrab3ss 3405

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 3134	. . 3 |- ( A C_ B <-> ( A i^i B ) = A )
2		ineq1 3321	. . . 4 |- ( ( A i^i B ) = A -> ( ( A i^i B ) i^i { x | ph } ) = ( A i^i { x | ph } ) )
3	2	eqcomd 2176	. . 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 3403	. 2 |- { x e. A | ph } = ( A i^i { x | ph } )
6		dfrab3 3403	. . . 4 |- { x e. B | ph } = ( B i^i { x | ph } )
7	6	ineq2i 3325	. . 3 |- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
8		inass 3337	. . 3 |- ( ( A i^i B ) i^i { x | ph } ) = ( A i^i ( B i^i { x | ph } ) )
9	7, 8	eqtr4i 2194	. 2 |- ( A i^i { x e. B | ph } ) = ( ( A i^i B ) i^i { x | ph } )
10	4, 5, 9	3eqtr4g 2228	1 |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Syntax hints:   -> wi 4   = wceq 1348   {cab 2156   {crab 2452   i^i cin 3120   C_ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134

This theorem is referenced by: (None)