Theorem oprabbid 3992

Description: Equivalent wff's yield equal operation class abstractions (deduction rule).

Hypotheses

Ref	Expression
oprabbid.1	|- (ph -> A.xph)
oprabbid.2	|- (ph -> A.yph)
oprabbid.3	|- (ph -> A.zph)
oprabbid.4	|- (ph -> (ps <-> ch))

Assertion

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

Distinct variable groups:   x,z   y,z

Proof of Theorem oprabbid

Step	Hyp	Ref	Expression
1		ax-17 970	. . 3 |- (ph -> A.wph)
2		oprabbid.3	. . 3 |- (ph -> A.zph)
3		oprabbid.1	. . . 4 |- (ph -> A.xph)
4		oprabbid.2	. . . . 5 |- (ph -> A.yph)
5		oprabbid.4	. . . . . 6 |- (ph -> (ps <-> ch))
6	5	anbi2d 615	. . . . 5 |- (ph -> ((w = <.x, y>. /\ ps) <-> (w = <.x, y>. /\ ch)))
7	4, 6	exbid 1104	. . . 4 |- (ph -> (E.y(w = <.x, y>. /\ ps) <-> E.y(w = <.x, y>. /\ ch)))
8	3, 7	exbid 1104	. . 3 |- (ph -> (E.xE.y(w = <.x, y>. /\ ps) <-> E.xE.y(w = <.x, y>. /\ ch)))
9	1, 2, 8	opabbid 2666	. 2 |- (ph -> {<.w, z>. | E.xE.y(w = <.x, y>. /\ ps)} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ch)})
10		dfoprab2 3988	. 2 |- {<.<.x, y>., z>. | ps} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ps)}
11		dfoprab2 3988	. 2 |- {<.<.x, y>., z>. | ch} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ch)}
12	9, 10, 11	3eqtr4g 1530	1 |- (ph -> {<.<.x, y>., z>. | ps} = {<.<.x, y>., z>. | ch})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955  E.wex 979  <.cop 2409  {copab 2663  {copab2 3961

This theorem is referenced by:  oprabbidv 3993  mapxpen 4488

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-opab 2664  df-oprab 3963

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