Theorem simplbiim 391

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)

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:  mpodifsnif  6181  ixpm  7012  finct  7456  apsscn  8976  zltaddlt1le  10412  pfxccatin12lem3  11506  oddnn02np1  12649  dvdsprmpweqnn  13117  sgrpass  13725  drnglring  14609  ausgrusgrben  16421