Theorem ssopab2i 4308

Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)

Hypothesis

Ref	Expression
ssopab2i.1	|- ( ph -> ps )

Assertion

Ref	Expression
ssopab2i	|- { <. x , y >. | ph } C_ { <. x , y >. | ps }

Proof of Theorem ssopab2i

Step	Hyp	Ref	Expression
1		ssopab2 4306	. 2 |- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )
2		ssopab2i.1	. . 3 |- ( ph -> ps )
3	2	ax-gen 1460	. 2 |- A. y ( ph -> ps )
4	1, 3	mpg 1462	1 |- { <. x , y >. | ph } C_ { <. x , y >. | ps }

Syntax hints:   -> wi 4   A.wal 1362   C_ wss 3153   {copab 4089

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3159  df-ss 3166  df-opab 4091

This theorem is referenced by:  brab2a  4712  opabssxp  4733  relopabiv  4785  funopab4  5291  ssoprab2i  6007  npsspw  7531  eqgfval  13292