Theorem elin3 3298

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 3290	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2	1	anbi1i 454	. 2 ⊢ ((𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷))
3		elin3.x	. . 3 ⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷)
4	3	elin2 3295	. 2 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷))
5		df-3an 965	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷))
6	2, 4, 5	3bitr4i 211	1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1335   ∈ wcel 2128   ∩ cin 3101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108

This theorem is referenced by: (None)