Theorem dfrab3ss 3441

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 3170	. . 3 |- ( A C_ B <-> ( A i^i B ) = A )
2		ineq1 3357	. . . 4 |- ( ( A i^i B ) = A -> ( ( A i^i B ) i^i { x | ph } ) = ( A i^i { x | ph } ) )
3	2	eqcomd 2202	. . 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 3439	. 2 |- { x e. A | ph } = ( A i^i { x | ph } )
6		dfrab3 3439	. . . 4 |- { x e. B | ph } = ( B i^i { x | ph } )
7	6	ineq2i 3361	. . 3 |- ( A i^i { x e. B | ph } ) = ( A i^i ( B i^i { x | ph } ) )
8		inass 3373	. . 3 |- ( ( A i^i B ) i^i { x | ph } ) = ( A i^i ( B i^i { x | ph } ) )
9	7, 8	eqtr4i 2220	. 2 |- ( A i^i { x e. B | ph } ) = ( ( A i^i B ) i^i { x | ph } )
10	4, 5, 9	3eqtr4g 2254	1 |- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )

Syntax hints:   -> wi 4   = wceq 1364   {cab 2182   {crab 2479   i^i cin 3156   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-in 3163  df-ss 3170

This theorem is referenced by: (None)