Theorem biimt 230

Description: A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)

Assertion

Ref	Expression
biimt	⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))

Proof of Theorem biimt

Step	Hyp	Ref	Expression
1		ax-1 5	. 2 ⊢ (𝜓 → (𝜑 → 𝜓))
2		pm2.27 35	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
3	1, 2	impbid2 131	1 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm5.5  231  a1bi  232  abai  494  dedlem0a  875  ceqsralt  2581  reu8  2737  csbiebt  2886  r19.3rm  3310  fncnv  4965  ovmpt2dxf  5626  brecop  6196