Theorem anabs1 783

Description: Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)

Assertion

Ref	Expression
anabs1	⊢ (((φ ∧ ψ) ∧ φ) ↔ (φ ∧ ψ))

Proof of Theorem anabs1

Step	Hyp	Ref	Expression
1		simpl 443	. . 3 ⊢ ((φ ∧ ψ) → φ)
2	1	pm4.71i 613	. 2 ⊢ ((φ ∧ ψ) ↔ ((φ ∧ ψ) ∧ φ))
3	2	bicomi 193	1 ⊢ (((φ ∧ ψ) ∧ φ) ↔ (φ ∧ ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358

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

This theorem is referenced by: (None)