Theorem jcab 527

Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)

Assertion

Ref	Expression
jcab	⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

Proof of Theorem jcab

Step	Hyp	Ref	Expression
1		simpl 488	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓)
2	1	imim2i 17	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))
3		simpr 490	. . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
4	3	imim2i 17	. . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒))
5	2, 4	jca 521	. 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))
6		pm3.43 479	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒)))
7	5, 6	impbii 212	1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)))

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

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

This theorem is used by:  pm4.76  528  pm5.44  552  ordi  1023  2mo2  2673  ssconb  4089  ssin  4184  2reu4lem  4479  tfr3  8400  trclfvcotr  15155  isprm2  16850  lgsquad2lem2  27705  ostthlem2  27948  pclclN  40928  ifpbibib  44495  elmapintrab  44561  elinintrab  44562  ismnuprim  45263