Theorem simp1bi 970

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
simp1bi	⊢ (φ → ψ)

Proof of Theorem simp1bi

Step	Hyp	Ref	Expression
1		3simp1bi.1	. . 3 ⊢ (φ ↔ (ψ ∧ χ ∧ θ))
2	1	biimpi 186	. 2 ⊢ (φ → (ψ ∧ χ ∧ θ))
3	2	simp1d 967	1 ⊢ (φ → ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by:  sfintfin  4533  sfin111  4537  vfinspsslem1  4551  weds  5939