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Theorem abbid 2727
 Description: Equivalent wff's yield equal class abstractions (deduction rule). (Contributed by NM, 21-Jun-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 2069 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 abbi 2724 . 2 (∀𝑥(𝜓𝜒) ↔ {𝑥𝜓} = {𝑥𝜒})
53, 4sylib 207 1 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  Ⅎwnf 1699  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606 This theorem is referenced by:  abbidv  2728  rabeqf  3165  sbcbid  3456  sbceqbidf  28705  opabdm  28803  opabrn  28804  fpwrelmap  28896  bj-rabbida2  32105  iotain  37640
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