Theorem anabs5 784

Description: Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)

Assertion

Ref	Expression
anabs5	⊢ ((φ ∧ (φ ∧ ψ)) ↔ (φ ∧ ψ))

Proof of Theorem anabs5

Step	Hyp	Ref	Expression
1		ibar 490	. . 3 ⊢ (φ → (ψ ↔ (φ ∧ ψ)))
2	1	bicomd 192	. 2 ⊢ (φ → ((φ ∧ ψ) ↔ ψ))
3	2	pm5.32i 618	1 ⊢ ((φ ∧ (φ ∧ ψ)) ↔ (φ ∧ ψ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358

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

This theorem is used by: (None)