Theorem jaob 963

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		pm2.67-2 891	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓))
2		olc 868	. . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	2	imim1i 63	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜒 → 𝜓))
4	1, 3	jca 511	. 2 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
5		pm3.44 961	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) → ((𝜑 ∨ 𝜒) → 𝜓))
6	4, 5	impbii 209	1 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847

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 207  df-an 396  df-or 848

This theorem is referenced by:  pm4.77  964  pm5.53  1006  pm4.83  1026  axio  2696  elunant  4194  intprg  4986  relop  5864  sqrt2irr  16282  algcvgblem  16611  efgred  19781  caucfil  25331  plydivex  26354  2sqlem6  27482  arg-ax  36399  tendoeq2  40757  ifpidg  43481