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Theorem abbidv 2137
Description: Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 10-Aug-1993.)
Hypothesis
Ref Expression
abbidv.1 (φ → (ψχ))
Assertion
Ref Expression
abbidv (φ → {xψ} = {xχ})
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)

Proof of Theorem abbidv
StepHypRef Expression
1 nfv 1402 . 2 xφ
2 abbidv.1 . 2 (φ → (ψχ))
31, 2abbid 2136 1 (φ → {xψ} = {xχ})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228  {cab 2008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015
This theorem is referenced by:  cdeqab  2731  csbeq1  2832  sbcel12g  2842  sbceqg  2843  csbeq2d  2851  csbnestgf  2875  csbprc  3239  ifbi  3325  pweq  3337  sneq  3361  csbsng  3405  rabsn  3411  dfopg  3521  opeq1  3523  opeq2  3524  csbunig  3562  unieq  3563  inteq  3592  iineq1  3645  iineq2  3648  dfiin2g  3664  iinrabm  3693  iinxprg  3705  opabbid  3796  csbxpg  4348  csbdmg  4456  imasng  4617  csbrng  4709  iotaeq  4802  iotabi  4803  dfimafn  5147  fnsnfv  5157  fndmin  5199  fliftf  5364  oprabbid  5481  recseq  5843  frec0g  5902  frecsuclem3  5906  frecsuc  5907  qseq1  6065  qseq2  6066  qsinxp  6093
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