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Theorem abbid 2151
Description: Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
abbid.1 xφ
abbid.2 (φ → (ψχ))
Assertion
Ref Expression
abbid (φ → {xψ} = {xχ})

Proof of Theorem abbid
StepHypRef Expression
1 abbid.1 . . 3 xφ
2 abbid.2 . . 3 (φ → (ψχ))
31, 2alrimi 1412 . 2 (φx(ψχ))
4 abbi 2148 . 2 (x(ψχ) ↔ {xψ} = {xχ})
53, 4sylib 127 1 (φ → {xψ} = {xχ})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  wnf 1346  {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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030
This theorem is referenced by:  abbidv  2152  rabeqf  2544  sbcbid  2810
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