Theorem opabbid 2674

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule).

Hypotheses

Ref	Expression
opabbid.1	|- (ph -> A.xph)
opabbid.2	|- (ph -> A.yph)
opabbid.3	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
opabbid	|- (ph -> {<.x, y>. | ps} = {<.x, y>. | ch})

Proof of Theorem opabbid

Step	Hyp	Ref	Expression
1		ax-17 973	. . 3 |- (ph -> A.zph)
2		opabbid.1	. . . 4 |- (ph -> A.xph)
3		opabbid.2	. . . . 5 |- (ph -> A.yph)
4		opabbid.3	. . . . . 6 |- (ph -> (ps <-> ch))
5	4	anbi2d 618	. . . . 5 |- (ph -> ((z = <.x, y>. /\ ps) <-> (z = <.x, y>. /\ ch)))
6	3, 5	exbid 1107	. . . 4 |- (ph -> (E.y(z = <.x, y>. /\ ps) <-> E.y(z = <.x, y>. /\ ch)))
7	2, 6	exbid 1107	. . 3 |- (ph -> (E.xE.y(z = <.x, y>. /\ ps) <-> E.xE.y(z = <.x, y>. /\ ch)))
8	1, 7	abbid 1579	. 2 |- (ph -> {z | E.xE.y(z = <.x, y>. /\ ps)} = {z | E.xE.y(z = <.x, y>. /\ ch)})
9		df-opab 2672	. 2 |- {<.x, y>. | ps} = {z | E.xE.y(z = <.x, y>. /\ ps)}
10		df-opab 2672	. 2 |- {<.x, y>. | ch} = {z | E.xE.y(z = <.x, y>. /\ ch)}
11	8, 9, 10	3eqtr4g 1534	1 |- (ph -> {<.x, y>. | ps} = {<.x, y>. | ch})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958  E.wex 982  {cab 1466  <.cop 2415  {copab 2671

This theorem is referenced by:  opabbidv 2675  oprabbid 4001  fnoprabg 4018  mapxpen 4501  xpmapenlem3 4504  xpmapenlem4 4505  xpmapenlem5 4506  sumeq2 6985  rnbra 10035

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-opab 2672

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