Theorem simplbda 379

Description: Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.)

Hypothesis

Ref	Expression
pm3.26bda.1	⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))

Assertion

Ref	Expression
simplbda	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Proof of Theorem simplbda

Step	Hyp	Ref	Expression
1		pm3.26bda.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))
2	1	biimpa 292	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃))
3	2	simprd 113	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  tgcl  12015  cldopn  12058  cncnp  12180  blgt0  12330  xblss2ps  12332  xblss2  12333  mopni  12410  metrest  12434  dvcl  12525