Theorem biimt 241

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 6	. 2 ⊢ (𝜓 → (𝜑 → 𝜓))
2		pm2.27 40	. 2 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
3	1, 2	impbid2 143	1 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  pm5.5  242  a1bi  243  abai  566  dedlem0a  981  ceqsralt  2849  reu8  3022  csbiebt  3187  r19.3rm  3616  fncnv  5447  ovmpodxf  6214  brecop  6899  tgss2  15182