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Theorem elrint2 4995

Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

**Assertion**
Ref	Expression
elrint2	⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Distinct variable groups: 𝑦,𝐵 𝑦,𝑋 Allowed substitution hint: 𝐴(𝑦)

**Proof of Theorem elrint2**
Step	Hyp	Ref	Expression
1		elrint 4994	. 2 ⊢ (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))
2	1	baib 535	1 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ (𝐴 ∩ ∩ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ 𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 ∀wral 3058 ∩ cin 3946 ∩ cint 4949

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699

This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-v 3473 df-in 3954 df-int 4950

This theorem is referenced by: mreacs 17637

	
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