Theorem ssoprab2i 5942

Description: Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypothesis

Ref	Expression
ssoprab2i.1	|- ( ph -> ps )

Assertion

Ref	Expression
ssoprab2i	|- { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps }

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)

Proof of Theorem ssoprab2i

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssoprab2i.1	. . . . 5 |- ( ph -> ps )
2	1	anim2i 340	. . . 4 |- ( ( w = <. x , y >. /\ ph ) -> ( w = <. x , y >. /\ ps ) )
3	2	2eximi 1594	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) -> E. x E. y ( w = <. x , y >. /\ ps ) )
4	3	ssopab2i 4262	. 2 |- { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } C_ { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ps ) }
5		dfoprab2 5900	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
6		dfoprab2 5900	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ps ) }
7	4, 5, 6	3sstr4i 3188	1 |- { <. <. x , y >. , z >. | ph } C_ { <. <. x , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   C_ wss 3121   <.cop 3586   {copab 4049   {coprab 5854

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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-oprab 5857

This theorem is referenced by: (None)