Theorem oel 441

Description: Elimination of redundant internal disjunct. Compare Theorem *4.45 of [WhiteheadRussell] p. 119.

Assertion

Ref	Expression
oel	⊢ (φ ↔ ((φ ∨ ψ) ∧ φ))

Proof of Theorem oel

Step	Hyp	Ref	Expression
1		orc 225	. . 3 ⊢ (φ → (φ ∨ ψ))
2	1	ancri 245	. 2 ⊢ (φ → ((φ ∨ ψ) ∧ φ))
3		pm3.27 260	. 2 ⊢ (((φ ∨ ψ) ∧ φ) → φ)
4	2, 3	impbi 139	1 ⊢ (φ ↔ ((φ ∨ ψ) ∧ φ))

Syntax hints:   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  prlem2 577

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-or 197  df-an 198

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