Theorem jaob 822

Description: Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)

Assertion

Ref	Expression
jaob	⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))

Proof of Theorem jaob

Step	Hyp	Ref	Expression
1		pm2.67-2 417	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓))
2		olc 399	. . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	2	imim1i 63	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜒 → 𝜓))
4	1, 3	jca 554	. 2 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
5		pm3.44 533	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) → ((𝜑 ∨ 𝜒) → 𝜓))
6	4, 5	impbii 199	1 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by:  pm4.77  828  pm5.53  837  pm4.83  970  axio  2592  unss  3787  ralunb  3794  intun  4509  intpr  4510  relop  5272  sqrt2irr  14979  algcvgblem  15290  efgred  18161  caucfil  23081  plydivex  24052  2sqlem6  25148  arg-ax  32415  tendoeq2  36062  ifpidg  37836