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Theorem abbid 2136
 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 1396 . 2 (φx(ψχ))
4 abbi 2133 . 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 1226   = wceq 1228  Ⅎwnf 1329  {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:  abbidv  2137  rabeqf  2528  sbcbid  2793
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