Theorem jaob 960

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		pm2.67-2 890	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓))
2		olc 866	. . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	2	imim1i 63	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜒 → 𝜓))
4	1, 3	jca 512	. 2 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
5		pm3.44 958	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) → ((𝜑 ∨ 𝜒) → 𝜓))
6	4, 5	impbii 208	1 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845

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 206  df-an 397  df-or 846

This theorem is referenced by:  pm4.77  961  pm5.53  1003  pm4.83  1023  axio  2693  elunant  4178  intprg  4985  intprOLD  4987  relop  5850  sqrt2irr  16194  algcvgblem  16516  efgred  19618  caucfil  24807  plydivex  25817  2sqlem6  26933  arg-ax  35393  tendoeq2  39737  ifpidg  42330