Theorem biimpac 292

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 157	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
3	2	imp 122	1 ⊢ ((𝜓 ∧ 𝜑) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  gencbvex2  2646  ordtri2or2exmidlem  4269  onsucelsucexmidlem  4272  ordsuc  4306  onsucuni2  4307  poltletr  4745  tz6.12-1  5221  nfunsn  5228  nnaordex  6123  th3qlem1  6231  ssfilem  6360  diffitest  6371  nqnq0pi  6628  distrlem1prl  6772  distrlem1pru  6773  eqle  7202  flodddiv4  10334