Theorem simp2bi 971

Description: Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
3simp1bi.1	⊢ (φ ↔ (ψ ∧ χ ∧ θ))

Assertion

Ref	Expression
simp2bi	⊢ (φ → χ)

Proof of Theorem simp2bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 ⊢ (φ ↔ (ψ ∧ χ ∧ θ))
2	1	biimpi 186	. 2 ⊢ (φ → (ψ ∧ χ ∧ θ))
3	2	simp2d 968	1 ⊢ (φ → χ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ w3a 934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by:  sfin111  4537  sbthlem3  6206