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Theorem abbid 2232
Description: Equivalent wff's yield equal class abstractions (deduction form). (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 1485 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 abbi 2229 . 2 (∀𝑥(𝜓𝜒) ↔ {𝑥𝜓} = {𝑥𝜒})
53, 4sylib 121 1 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312   = wceq 1314  wnf 1419  {cab 2101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108
This theorem is referenced by:  abbidv  2233  rabeqf  2648  sbcbid  2936
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