Theorem elunant 4137

Description: A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)

Assertion

Ref	Expression
elunant	⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑)))

Proof of Theorem elunant

Step	Hyp	Ref	Expression
1		elun 4107	. . 3 ⊢ (𝐶 ∈ (𝐴 ∪ 𝐵) ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵))
2	1	imbi1i 352	. 2 ⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵) → 𝜑))
3		jaob 976	. 2 ⊢ (((𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑)))
4	2, 3	bitri 278	1 ⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∈ wcel 2146   ∪ cun 3904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911

This theorem is used by:  unss  4143  ralunb  4150  srcmpltd  4439  intun  4947