Theorem anabs5 507

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 285	. . 3 ⊢ (φ → (ψ ↔ (φ ∧ ψ)))
2	1	bicomd 129	. 2 ⊢ (φ → ((φ ∧ ψ) ↔ ψ))
3	2	pm5.32i 427	1 ⊢ ((φ ∧ (φ ∧ ψ)) ↔ (φ ∧ ψ))

Syntax hints:   ∧ wa 97   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  mo3h  1950  indif  3174  axsep2  3867