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Theorem abbi2dv 2138
Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
Hypothesis
Ref Expression
abbirdv.1 (φ → (x Aψ))
Assertion
Ref Expression
abbi2dv (φA = {xψ})
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem abbi2dv
StepHypRef Expression
1 abbirdv.1 . . 3 (φ → (x Aψ))
21alrimiv 1736 . 2 (φx(x Aψ))
3 abeq2 2128 . 2 (A = {xψ} ↔ x(x Aψ))
42, 3sylibr 137 1 (φA = {xψ})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228   wcel 1374  {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-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018
This theorem is referenced by:  sbab  2146  iftrue  3315  iffalse  3317  iniseg  4624  fncnvima2  5213  isoini  5382  dftpos3  5799
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