Theorem biimpac 298

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 159	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	imp 124	1 ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105

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

This proof depends on definitions:  df-bi 117

This theorem is used by:  gencbvex2  2870  sseq0  3565  ordtri2or2exmidlem  4673  onsucelsucexmidlem  4676  ordsuc  4710  onsucuni2  4711  poltletr  5188  tz6.12-1  5722  nfunsn  5733  nnaordex  6801  th3qlem1  6911  ssfilem  7177  ssfilemd  7179  diffitest  7191  nqnq0pi  7805  distrlem1prl  7949  distrlem1pru  7950  eqle  8417  swrd0g  11432  flodddiv4  12703  zabsle1  16118