Theorem simplbiim 385

Description: Implication from an eliminated conjunct equivalent to the antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
simplbiim.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
simplbiim.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
simplbiim	⊢ (𝜑 → 𝜃)

Proof of Theorem simplbiim

Step	Hyp	Ref	Expression
1		simplbiim.1	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2		simplbiim.2	. . 3 ⊢ (𝜒 → 𝜃)
3	2	adantl 275	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4	1, 3	sylbi 120	1 ⊢ (𝜑 → 𝜃)

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  mpodifsnif  5946  ixpm  6708  finct  7093  apsscn  8566  zltaddlt1le  9964  oddnn02np1  11839  dvdsprmpweqnn  12289  sgrpass  12649