Theorem elin3 4106

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 4096	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2	1	anbi1i 636	. 2 ⊢ ((𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷))
3		elin3.x	. . 3 ⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷)
4	3	elin2 4103	. 2 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷))
5		df-3an 1105	. 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))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ∩ cin 4092

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094

This theorem is used by: (None)