Theorem ineqri 3450

Description: Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
ineqri.1	⊢ ((x ∈ A ∧ x ∈ B) ↔ x ∈ C)

Assertion

Ref	Expression
ineqri	⊢ (A ∩ B) = C

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem ineqri

Step	Hyp	Ref	Expression
1		elin 3220	. . 3 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
2		ineqri.1	. . 3 ⊢ ((x ∈ A ∧ x ∈ B) ↔ x ∈ C)
3	1, 2	bitri 240	. 2 ⊢ (x ∈ (A ∩ B) ↔ x ∈ C)
4	3	eqriv 2350	1 ⊢ (A ∩ B) = C

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by:  inidm  3465  inass  3466  dfin2  3492  indi  3502  inab  3523  in0  3577  dmres  4987