Theorem stbid 822

Description: The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)

Hypothesis

Ref	Expression
stbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
stbid	⊢ (𝜑 → (STAB 𝜓 ↔ STAB 𝜒))

Proof of Theorem stbid

Step	Hyp	Ref	Expression
1		stbid.1	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	notbid 657	. . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒))
3	2	notbid 657	. . 3 ⊢ (𝜑 → (¬ ¬ 𝜓 ↔ ¬ ¬ 𝜒))
4	3, 1	imbi12d 233	. 2 ⊢ (𝜑 → ((¬ ¬ 𝜓 → 𝜓) ↔ (¬ ¬ 𝜒 → 𝜒)))
5		df-stab 821	. 2 ⊢ (STAB 𝜓 ↔ (¬ ¬ 𝜓 → 𝜓))
6		df-stab 821	. 2 ⊢ (STAB 𝜒 ↔ (¬ ¬ 𝜒 → 𝜒))
7	4, 5, 6	3bitr4g 222	1 ⊢ (𝜑 → (STAB 𝜓 ↔ STAB 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  STAB wstab 820

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 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-stab 821

This theorem is referenced by:  dfss4st  3355  cnstab  8543  exmid1stab  13880