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Theorem abbid 2170
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 1431 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 abbi 2167 . 2 (∀𝑥(𝜓𝜒) ↔ {𝑥𝜓} = {𝑥𝜒})
53, 4sylib 131 1 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wnf 1365  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049
This theorem is referenced by:  abbidv  2171  rabeqf  2567  sbcbid  2843
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