Theorem pm5.54dc 919

Description: A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)

Assertion

Ref	Expression
pm5.54dc	⊢ (DECID 𝜑 → (((𝜑 ∧ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∧ 𝜓) ↔ 𝜓)))

Proof of Theorem pm5.54dc

Step	Hyp	Ref	Expression
1		df-dc 836	. . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
3		ax-ia3 108	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
4	2, 3	impbid2 143	. . . 4 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜓))
5		simpl 109	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
6		ax-in2 616	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → (𝜑 ∧ 𝜓)))
7	5, 6	impbid2 143	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∧ 𝜓) ↔ 𝜑))
8	4, 7	orim12i 760	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 ∧ 𝜓) ↔ 𝜓) ∨ ((𝜑 ∧ 𝜓) ↔ 𝜑)))
9	1, 8	sylbi 121	. 2 ⊢ (DECID 𝜑 → (((𝜑 ∧ 𝜓) ↔ 𝜓) ∨ ((𝜑 ∧ 𝜓) ↔ 𝜑)))
10	9	orcomd 730	1 ⊢ (DECID 𝜑 → (((𝜑 ∧ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∧ 𝜓) ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by: (None)