Theorem simplbda 384

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 296	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃))
3	2	simprd 114	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

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

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

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  rhmpropd  13886  lsslmod  14012  tgcl  14384  cldopn  14427  cncnp  14550  blgt0  14722  xblss2ps  14724  xblss2  14725  mopni  14802  metrest  14826  dvcl  15003  dvcnp2cntop  15019