Theorem elin3 3363

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 3355	. . 3 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2	1	anbi1i 458	. 2 ⊢ ((𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷))
3		elin3.x	. . 3 ⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷)
4	3	elin2 3360	. 2 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ (𝐵 ∩ 𝐶) ∧ 𝐴 ∈ 𝐷))
5		df-3an 982	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷) ↔ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) ∧ 𝐴 ∈ 𝐷))
6	2, 4, 5	3bitr4i 212	1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1372   ∈ wcel 2175   ∩ cin 3164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171

This theorem is referenced by: (None)