Theorem ssopab2 4234

Description: Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)

Assertion

Ref	Expression
ssopab2	|- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )

Proof of Theorem ssopab2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1521	. . . 4 |- F/ x A. x A. y ( ph -> ps )
2		nfa1 1521	. . . . . 6 |- F/ y A. y ( ph -> ps )
3		sp 1491	. . . . . . 7 |- ( A. y ( ph -> ps ) -> ( ph -> ps ) )
4	3	anim2d 335	. . . . . 6 |- ( A. y ( ph -> ps ) -> ( ( z = <. x , y >. /\ ph ) -> ( z = <. x , y >. /\ ps ) ) )
5	2, 4	eximd 1592	. . . . 5 |- ( A. y ( ph -> ps ) -> ( E. y ( z = <. x , y >. /\ ph ) -> E. y ( z = <. x , y >. /\ ps ) ) )
6	5	sps 1517	. . . 4 |- ( A. x A. y ( ph -> ps ) -> ( E. y ( z = <. x , y >. /\ ph ) -> E. y ( z = <. x , y >. /\ ps ) ) )
7	1, 6	eximd 1592	. . 3 |- ( A. x A. y ( ph -> ps ) -> ( E. x E. y ( z = <. x , y >. /\ ph ) -> E. x E. y ( z = <. x , y >. /\ ps ) ) )
8	7	ss2abdv 3201	. 2 |- ( A. x A. y ( ph -> ps ) -> { z | E. x E. y ( z = <. x , y >. /\ ph ) } C_ { z | E. x E. y ( z = <. x , y >. /\ ps ) } )
9		df-opab 4026	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
10		df-opab 4026	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
11	8, 9, 10	3sstr4g 3171	1 |- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1333   = wceq 1335   E.wex 1472   {cab 2143   C_ wss 3102   <.cop 3563   {copab 4024

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-opab 4026

This theorem is referenced by:  ssopab2b  4235  ssopab2i  4236  ssopab2dv  4237