Theorem oprabbidv 3993

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

Hypothesis

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

Assertion

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

Distinct variable groups:   x,z,ph   y,z,ph

Proof of Theorem oprabbidv

Step	Hyp	Ref	Expression
1		ax-17 970	. 2 |- (ph -> A.xph)
2		ax-17 970	. 2 |- (ph -> A.yph)
3		ax-17 970	. 2 |- (ph -> A.zph)
4		oprabbidv.1	. 2 |- (ph -> (ps <-> ch))
5	1, 2, 3, 4	oprabbid 3992	1 |- (ph -> {<.<.x, y>., z>. | ps} = {<.<.x, y>., z>. | ch})

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955  {copab2 3961

This theorem is referenced by:  oprabbii 3994  blfval 7816  grpdivfval 8064  nvmfval 8249  ipfval 8338  symgval 10394  ishoma 10666  ishomb 10667

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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