Theorem simplbiim 387

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 277	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
4	1, 3	sylbi 121	1 ⊢ (𝜑 → 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  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mpodifsnif  5971  ixpm  6733  finct  7118  apsscn  8607  zltaddlt1le  10010  oddnn02np1  11888  dvdsprmpweqnn  12338  sgrpass  12820