Theorem biimpac 294

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-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  gencbvex2  2702  ordtri2or2exmidlem  4399  onsucelsucexmidlem  4402  ordsuc  4436  onsucuni2  4437  poltletr  4895  tz6.12-1  5400  nfunsn  5407  nnaordex  6374  th3qlem1  6482  ssfilem  6719  diffitest  6731  nqnq0pi  7187  distrlem1prl  7331  distrlem1pru  7332  eqle  7771  flodddiv4  11472