Theorem ssopab2 5001

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		id 22	. . . . . 6 |- ( ( ph -> ps ) -> ( ph -> ps ) )
2	1	anim2d 589	. . . . 5 |- ( ( ph -> ps ) -> ( ( z = <. x , y >. /\ ph ) -> ( z = <. x , y >. /\ ps ) ) )
3	2	aleximi 1759	. . . 4 |- ( A. y ( ph -> ps ) -> ( E. y ( z = <. x , y >. /\ ph ) -> E. y ( z = <. x , y >. /\ ps ) ) )
4	3	aleximi 1759	. . 3 |- ( A. x A. y ( ph -> ps ) -> ( E. x E. y ( z = <. x , y >. /\ ph ) -> E. x E. y ( z = <. x , y >. /\ ps ) ) )
5	4	ss2abdv 3675	. 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 ) } )
6		df-opab 4713	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
7		df-opab 4713	. 2 |- { <. x , y >. | ps } = { z | E. x E. y ( z = <. x , y >. /\ ps ) }
8	5, 6, 7	3sstr4g 3646	1 |- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   {cab 2608   C_ wss 3574   <.cop 4183   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-opab 4713

This theorem is referenced by:  ssopab2b  5002  ssopab2i  5003  ssopab2dv  5004  opabbrex  6695