Theorem pm4.45im 332

Description: Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
pm4.45im	⊢ (𝜑 ↔ (𝜑 ∧ (𝜓 → 𝜑)))

Proof of Theorem pm4.45im

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (𝜑 → (𝜓 → 𝜑))
2	1	ancli 321	. 2 ⊢ (𝜑 → (𝜑 ∧ (𝜓 → 𝜑)))
3		simpl 108	. 2 ⊢ ((𝜑 ∧ (𝜓 → 𝜑)) → 𝜑)
4	2, 3	impbii 125	1 ⊢ (𝜑 ↔ (𝜑 ∧ (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  difdif  3247