Theorem elin3 4100

Description: Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypothesis

Ref	Expression
elin3.x	⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷)

Assertion

Ref	Expression
elin3	⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))

Proof of Theorem elin3

Step	Hyp	Ref	Expression
1		elin 3869	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2	1	anbi1i 627	. 2 ⊢ ((𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷))
3		elin3.x	. . 3 ⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷)
4	3	elin2 4097	. 2 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷))
5		df-3an 1091	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷))
6	2, 4, 5	3bitr4i 306	1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ∩ cin 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-in 3860

This theorem is referenced by: (None)