Theorem biimpac 296

Description: Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)

Hypothesis

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

Assertion

Ref	Expression
biimpac	⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Proof of Theorem biimpac

Step	Hyp	Ref	Expression
1		biimpa.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	biimpcd 158	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	imp 123	1 ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  gencbvex2  2773  ordtri2or2exmidlem  4503  onsucelsucexmidlem  4506  ordsuc  4540  onsucuni2  4541  poltletr  5004  tz6.12-1  5513  nfunsn  5520  nnaordex  6495  th3qlem1  6603  ssfilem  6841  diffitest  6853  nqnq0pi  7379  distrlem1prl  7523  distrlem1pru  7524  eqle  7990  flodddiv4  11871  zabsle1  13550