Theorem ssopab2i 4715

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 4713	. 2 |- (A.xA.y(ph -> ps) -> {<.x, y>. | ph} C_ {<.x, y>. | ps})
2		ssopab2i.1	. . 3 |- (ph -> ps)
3	2	ax-gen 1546	. 2 |- A.y(ph -> ps)
4	1, 3	mpg 1548	1 |- {<.x, y>. | ph} C_ {<.x, y>. | ps}

Syntax hints:   -> wi 4  A.wal 1540   C_ wss 3258  {copab 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624

This theorem is referenced by:  opabssxp  4838  funopab4  5142  ssoprab2i  5581