Theorem exbiri 382

Description: Inference form of exbir 1486. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.)

Hypothesis

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

Assertion

Ref	Expression
exbiri	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

Proof of Theorem exbiri

Step	Hyp	Ref	Expression
1		exbiri.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
2	1	biimpar 297	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜃) → 𝜒)
3	2	exp31 364	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

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

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

This theorem is used by:  biimp3ar  1387  eqrdav  2237  tfrlem9  6594  mapfset  6949  sbthlem1  7280  lbreu  9284  uzsubsubfz  10470  elfzodifsumelfzo  10637  pfxccatin12lem3  11527  cncfmptid  15835  addccncf  15838  negcncf  15843  gausslemma2dlem1a  16394  usgredg2vlem2  16681  clwwlkccatlem  16858