Theorem jaob 976

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		pm2.67-2 905	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓))
2		olc 882	. . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒))
3	2	imim1i 64	. . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜒 → 𝜓))
4	1, 3	jca 521	. 2 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))
5		pm3.44 974	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) → ((𝜑 ∨ 𝜒) → 𝜓))
6	4, 5	impbii 212	1 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  pm4.77  977  pm5.53  1022  pm4.83  1042  axio  2723  elunant  4130  intprg  4941  relop  5828  sqrt2irr  16417  algcvgblem  16752  efgred  19962  caucfil  25604  plydivex  26618  2sqlem6  27750  arg-ax  37204  mh-prprimbi  37331  tendoeq2  41831  ifpidg  44491