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Theorem stbid 817
Description: The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)
Hypothesis
Ref Expression
stbid.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
stbid (𝜑 → (STAB 𝜓STAB 𝜒))

Proof of Theorem stbid
StepHypRef Expression
1 stbid.1 . . . . 5 (𝜑 → (𝜓𝜒))
21notbid 656 . . . 4 (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
32notbid 656 . . 3 (𝜑 → (¬ ¬ 𝜓 ↔ ¬ ¬ 𝜒))
43, 1imbi12d 233 . 2 (𝜑 → ((¬ ¬ 𝜓𝜓) ↔ (¬ ¬ 𝜒𝜒)))
5 df-stab 816 . 2 (STAB 𝜓 ↔ (¬ ¬ 𝜓𝜓))
6 df-stab 816 . 2 (STAB 𝜒 ↔ (¬ ¬ 𝜒𝜒))
74, 5, 63bitr4g 222 1 (𝜑 → (STAB 𝜓STAB 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  STAB wstab 815
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-in1 603  ax-in2 604
This theorem depends on definitions:  df-bi 116  df-stab 816
This theorem is referenced by:  dfss4st  3309  exmid1stab  13279
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