Theorem ssopab2i 4342

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 4340	. 2 |- ( A. x A. y ( ph -> ps ) -> { <. x , y >. | ph } C_ { <. x , y >. | ps } )
2		ssopab2i.1	. . 3 |- ( ph -> ps )
3	2	ax-gen 1473	. 2 |- A. y ( ph -> ps )
4	1, 3	mpg 1475	1 |- { <. x , y >. | ph } C_ { <. x , y >. | ps }

Syntax hints:   -> wi 4   A.wal 1371   C_ wss 3174   {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:  brab2a  4746  opabssxp  4767  relopabiv  4819  funopab4  5327  ssoprab2i  6057  npsspw  7619  eqgfval  13673