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)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  gencbvex2  2848  ordtri2or2exmidlem  4617  onsucelsucexmidlem  4620  ordsuc  4654  onsucuni2  4655  poltletr  5128  tz6.12-1  5653  nfunsn  5663  nnaordex  6672  th3qlem1  6782  ssfilem  7033  diffitest  7045  nqnq0pi  7621  distrlem1prl  7765  distrlem1pru  7766  eqle  8234  swrd0g  11187  flodddiv4  12442  zabsle1  15672