Theorem pm4.45im 545

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 534	. 2 ⊢ (φ → (φ ∧ (ψ → φ)))
3		simpl 443	. 2 ⊢ ((φ ∧ (ψ → φ)) → φ)
4	2, 3	impbii 180	1 ⊢ (φ ↔ (φ ∧ (ψ → φ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  difdif  3392