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  4675  onsucelsucexmidlem  4678  ordsuc  4712  onsucuni2  4713  poltletr  5191  tz6.12-1  5725  nfunsn  5736  nnaordex  6805  th3qlem1  6915  ssfilem  7181  ssfilemd  7183  diffitest  7195  nqnq0pi  7811  distrlem1prl  7955  distrlem1pru  7956  eqle  8423  swrd0g  11454  flodddiv4  12728  zabsle1  16323