Theorem pm4.79dc 808

Description: Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)

Assertion

Ref	Expression
pm4.79dc	⊢ (DECID φ → (DECID ψ → (((ψ → φ) ∨ (χ → φ)) ↔ ((ψ ∧ χ) → φ))))

Proof of Theorem pm4.79dc

Step	Hyp	Ref	Expression
1		id 19	. . . 4 ⊢ ((ψ → φ) → (ψ → φ))
2		id 19	. . . 4 ⊢ ((χ → φ) → (χ → φ))
3	1, 2	jaoa 639	. . 3 ⊢ (((ψ → φ) ∨ (χ → φ)) → ((ψ ∧ χ) → φ))
4		simplimdc 756	. . . . . 6 ⊢ (DECID ψ → (¬ (ψ → φ) → ψ))
5		pm3.3 248	. . . . . 6 ⊢ (((ψ ∧ χ) → φ) → (ψ → (χ → φ)))
6	4, 5	syl9 66	. . . . 5 ⊢ (DECID ψ → (((ψ ∧ χ) → φ) → (¬ (ψ → φ) → (χ → φ))))
7		dcim 783	. . . . . 6 ⊢ (DECID ψ → (DECID φ → DECID (ψ → φ)))
8		pm2.54dc 789	. . . . . 6 ⊢ (DECID (ψ → φ) → ((¬ (ψ → φ) → (χ → φ)) → ((ψ → φ) ∨ (χ → φ))))
9	7, 8	syl6 29	. . . . 5 ⊢ (DECID ψ → (DECID φ → ((¬ (ψ → φ) → (χ → φ)) → ((ψ → φ) ∨ (χ → φ)))))
10	6, 9	syl5d 62	. . . 4 ⊢ (DECID ψ → (DECID φ → (((ψ ∧ χ) → φ) → ((ψ → φ) ∨ (χ → φ)))))
11	10	imp 115	. . 3 ⊢ ((DECID ψ ∧ DECID φ) → (((ψ ∧ χ) → φ) → ((ψ → φ) ∨ (χ → φ))))
12	3, 11	impbid2 131	. 2 ⊢ ((DECID ψ ∧ DECID φ) → (((ψ → φ) ∨ (χ → φ)) ↔ ((ψ ∧ χ) → φ)))
13	12	expcom 109	1 ⊢ (DECID φ → (DECID ψ → (((ψ → φ) ∨ (χ → φ)) ↔ ((ψ ∧ χ) → φ))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  DECID wdc 741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by: (None)