Theorem ssopab2 4370

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 1589	. . . 4 |- F/ x A. x A. y ( ph -> ps )
2		nfa1 1589	. . . . . 6 |- F/ y A. y ( ph -> ps )
3		sp 1559	. . . . . . 7 |- ( A. y ( ph -> ps ) -> ( ph -> ps ) )
4	3	anim2d 337	. . . . . 6 |- ( A. y ( ph -> ps ) -> ( ( z = <. x , y >. /\ ph ) -> ( z = <. x , y >. /\ ps ) ) )
5	2, 4	eximd 1660	. . . . 5 |- ( A. y ( ph -> ps ) -> ( E. y ( z = <. x , y >. /\ ph ) -> E. y ( z = <. x , y >. /\ ps ) ) )
6	5	sps 1585	. . . 4 |- ( A. x A. y ( ph -> ps ) -> ( E. y ( z = <. x , y >. /\ ph ) -> E. y ( z = <. x , y >. /\ ps ) ) )
7	1, 6	eximd 1660	. . 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 3300	. 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 4151	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
10		df-opab 4151	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
11	8, 9, 10	3sstr4g 3270	1 |- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1395   = wceq 1397   E.wex 1540   {cab 2217   C_ wss 3200   <.cop 3672   {copab 4149

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-opab 4151

This theorem is referenced by:  ssopab2b  4371  ssopab2i  4372  ssopab2dv  4373