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Theorem abbi2dv 2153
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 1751 . 2 (φx(x Aψ))
3 abeq2 2143 . 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 1240   = wceq 1242   wcel 1390  {cab 2023
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033
This theorem is referenced by:  sbab  2161  iftrue  3330  iffalse  3333  iniseg  4640  fncnvima2  5231  isoini  5400  dftpos3  5818
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