Theorem ssopab2 4713

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.xA.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 1788	. . . 4 |- F/xA.xA.y(ph -> ps)
2		nfa1 1788	. . . . . 6 |- F/yA.y(ph -> ps)
3		sp 1747	. . . . . . 7 |- (A.y(ph -> ps) -> (ph -> ps))
4	3	anim2d 548	. . . . . 6 |- (A.y(ph -> ps) -> ((z = <.x, y>. /\ ph) -> (z = <.x, y>. /\ ps)))
5	2, 4	eximd 1770	. . . . 5 |- (A.y(ph -> ps) -> (E.y(z = <.x, y>. /\ ph) -> E.y(z = <.x, y>. /\ ps)))
6	5	sps 1754	. . . 4 |- (A.xA.y(ph -> ps) -> (E.y(z = <.x, y>. /\ ph) -> E.y(z = <.x, y>. /\ ps)))
7	1, 6	eximd 1770	. . 3 |- (A.xA.y(ph -> ps) -> (E.xE.y(z = <.x, y>. /\ ph) -> E.xE.y(z = <.x, y>. /\ ps)))
8	7	ss2abdv 3340	. 2 |- (A.xA.y(ph -> ps) -> {z | E.xE.y(z = <.x, y>. /\ ph)} C_ {z | E.xE.y(z = <.x, y>. /\ ps)})
9		df-opab 4624	. 2 |- {<.x, y>. | ph} = {z | E.xE.y(z = <.x, y>. /\ ph)}
10		df-opab 4624	. 2 |- {<.x, y>. | ps} = {z | E.xE.y(z = <.x, y>. /\ ps)}
11	8, 9, 10	3sstr4g 3313	1 |- (A.xA.y(ph -> ps) -> {<.x, y>. | ph} C_ {<.x, y>. | ps})

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  {cab 2339   C_ wss 3258  <.cop 4562  {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:  ssopab2b  4714  ssopab2i  4715  ssopab2dv  4716