Theorem anabs1 374

Description: Absorption into embedded conjunct.

Assertion

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

Proof of Theorem anabs1

Step	Hyp	Ref	Expression
1		pm3.26 256	. 2 ⊢ (((φ ∧ ψ) ∧ φ) → (φ ∧ ψ))
2		id 9	. . 3 ⊢ ((φ ∧ ψ) → (φ ∧ ψ))
3		pm3.26 256	. . 3 ⊢ ((φ ∧ ψ) → φ)
4	2, 3	jca 236	. 2 ⊢ ((φ ∧ ψ) → ((φ ∧ ψ) ∧ φ))
5	1, 4	impbi 139	1 ⊢ (((φ ∧ ψ) ∧ φ) ↔ (φ ∧ ψ))

Syntax hints:   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  anabs5 375  euanv 1053  poirr 2133

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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