Theorem bioprabi 3027

Description: Equivalent wff's yield equal operation class abstractions.

Hypothesis

Ref	Expression
bioprab.1	|- (ph <-> ps)

Assertion

Ref	Expression
bioprabi	|- {<.<.x, y>., z>. | ph} = {<.<.x, y>., z>. | ps}

Distinct variable group(s):   x,y,z

Proof of Theorem bioprabi

Step	Hyp	Ref	Expression
1		cleqid 1102	. 2 |- w = w
2		bioprab.1	. . . 4 |- (ph <-> ps)
3	2	a1i 7	. . 3 |- (w = w -> (ph <-> ps))
4	3	bioprabdv 3026	. 2 |- (w = w -> {<.<.x, y>., z>. | ph} = {<.<.x, y>., z>. | ps})
5	1, 4	ax-mp 6	1 |- {<.<.x, y>., z>. | ph} = {<.<.x, y>., z>. | ps}

Syntax hints:   <-> wb 127   = weq 797   = wceq 1091  {copab2 3002

This theorem is referenced by:  1st2val 3097  df1st2 3098  oprec 3254  fnmap 3262  mapvalg 3263  cdavalt 3716  addcnsr 4047  mulcnsr 4048  axaddex 4059  axmulex 4060  seqval 4665  ruclem13 4897  sshjvalt 5321  dfchj2 5325  dfchj3 5326  sshjval3t 5327  hosmvalt 5487  hodmvalt 5488

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-oprab 3004

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