Theorem dfrab3ss 3534

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 3260	. . 3 |- (A C_ B <-> (A i^i B) = A)
2		ineq1 3451	. . . 4 |- ((A i^i B) = A -> ((A i^i B) i^i {x | ph}) = (A i^i {x | ph}))
3	2	eqcomd 2358	. . 3 |- ((A i^i B) = A -> (A i^i {x | ph}) = ((A i^i B) i^i {x | ph}))
4	1, 3	sylbi 187	. 2 |- (A C_ B -> (A i^i {x | ph}) = ((A i^i B) i^i {x | ph}))
5		dfrab3 3532	. 2 |- {x e. A | ph} = (A i^i {x | ph})
6		dfrab3 3532	. . . 4 |- {x e. B | ph} = (B i^i {x | ph})
7	6	ineq2i 3455	. . 3 |- (A i^i {x e. B | ph}) = (A i^i (B i^i {x | ph}))
8		inass 3466	. . 3 |- ((A i^i B) i^i {x | ph}) = (A i^i (B i^i {x | ph}))
9	7, 8	eqtr4i 2376	. 2 |- (A i^i {x e. B | ph}) = ((A i^i B) i^i {x | ph})
10	4, 5, 9	3eqtr4g 2410	1 |- (A C_ B -> {x e. A | ph} = (A i^i {x e. B | ph}))

Syntax hints:   -> wi 4   = wceq 1642  {cab 2339  {crab 2619   i^i cin 3209   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)