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Theorem abbid 2466
 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 → {x ψ} = {x χ})

Proof of Theorem abbid
StepHypRef Expression
1 abbid.1 . . 3 xφ
2 abbid.2 . . 3 (φ → (ψχ))
31, 2alrimi 1765 . 2 (φx(ψχ))
4 abbi 2463 . 2 (x(ψχ) ↔ {x ψ} = {x χ})
53, 4sylib 188 1 (φ → {x ψ} = {x χ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  {cab 2339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346 This theorem is referenced by:  abbidv  2467  rabeqf  2852  sbcbid  3099
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