Theorem biimp3ar 1383

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 1220	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1007

This theorem is referenced by:  rmoi  3140  brelrng  4995  nn0p1elfzo  10548  ssfzo12  10596  abssubge0  11818  qredeu  12825  basgen2  15077  logbprmirr  15968  lgssq  16044  lgssq2  16045  usgr0v  16363