Theorem ssopab2 4340

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 1565	. . . 4 |- F/ x A. x A. y ( ph -> ps )
2		nfa1 1565	. . . . . 6 |- F/ y A. y ( ph -> ps )
3		sp 1535	. . . . . . 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 1636	. . . . 5 |- ( A. y ( ph -> ps ) -> ( E. y ( z = <. x , y >. /\ ph ) -> E. y ( z = <. x , y >. /\ ps ) ) )
6	5	sps 1561	. . . 4 |- ( A. x A. y ( ph -> ps ) -> ( E. y ( z = <. x , y >. /\ ph ) -> E. y ( z = <. x , y >. /\ ps ) ) )
7	1, 6	eximd 1636	. . 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 3274	. 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 4122	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
10		df-opab 4122	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
11	8, 9, 10	3sstr4g 3244	1 |- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   E.wex 1516   {cab 2193   C_ wss 3174   <.cop 3646   {copab 4120

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187  df-opab 4122

This theorem is referenced by:  ssopab2b  4341  ssopab2i  4342  ssopab2dv  4343