Theorem pm4.83 1021

Description: Theorem *4.83 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.83	⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓)

Proof of Theorem pm4.83

Step	Hyp	Ref	Expression
1		exmid 891	. . 3 ⊢ (𝜑 ∨ ¬ 𝜑)
2	1	a1bi 365	. 2 ⊢ (𝜓 ↔ ((𝜑 ∨ ¬ 𝜑) → 𝜓))
3		jaob 958	. 2 ⊢ (((𝜑 ∨ ¬ 𝜑) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)))
4	2, 3	bitr2i 278	1 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜓)) ↔ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843

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 209  df-an 399  df-or 844

This theorem is referenced by:  cases2  1042  dmdbr5ati  30198  cvlsupr3  36479  rp-fakeanorass  39877