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Theorem abbid 2795
Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) Avoid ax-10 2129 and ax-11 2146. (Revised by Wolf Lammen, 6-May-2023.)
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 2198 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 abbi 2792 . 2 (∀𝑥(𝜓𝜒) → {𝑥𝜓} = {𝑥𝜒})
53, 4syl 17 1 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wnf 1777  {cab 2701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716
This theorem is referenced by:  rabbida4  3449  rabeqf  3458  rabeqiOLD  3463  sbcbid  3828  sbceqbidf  32222  opabdm  32335  opabrn  32336  fpwrelmap  32453  sticksstones16  41513  rabbida2  44370  rabbida3  44373
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