Theorem biimp3ar 1387

Description: Infer implication from a logical equivalence. Similar to biimpar 297. (Contributed by NM, 2-Jan-2009.)

Hypothesis

Ref	Expression
biimp3a.1	⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
biimp3ar	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)

Proof of Theorem biimp3ar

Step	Hyp	Ref	Expression
1		biimp3a.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
2	1	exbiri 382	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
3	2	3imp 1224	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  rmoi  3146  brelrng  5013  nn0p1elfzo  10596  ssfzo12  10644  abssubge0  11870  qredeu  12877  basgen2  15184  logbprmirr  16080  lgssq  16171  lgssq2  16172  usgr0v  16490