Theorem biimp3ar 1471

Description: Infer implication from a logical equivalence. Similar to biimpar 479. (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 810	. 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))
3	2	3imp 1112	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090

This theorem is referenced by:  rmoi  3886  brelrng  5941  fpr3g  8270  frrlem4  8274  dif1enlem  9156  dif1enlemOLD  9157  php3  9212  div2sub  12039  nn0p1elfzo  13675  ssfzo12  13725  modltm1p1mod  13888  hashgt23el  14384  repswpfx  14735  abssubge0  15274  qredeu  16595  abvne0  20435  slesolinvbi  22183  basgen2  22492  fcfval  23537  nmne0  24128  ovolfsf  24988  logbprmirr  26301  lgssq  26840  lgssq2  26841  colinearalg  28168  usgr0v  28498  frgr0vb  29516  nv1  29928  adjeq  31188  pridln1  32561  revpfxsfxrev  34106  areacirc  36581  fvopabf4g  36590  exidreslem  36745  hgmapvvlem3  40796  iocmbl  41962  iunconnlem2  43696  ssfz12  46022  m1modmmod  47207