Theorem ineqri 4211

Description: Inference from membership to intersection. (Contributed by NM, 21-Jun-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 3966	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		ineqri.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)
3	1, 2	bitri 275	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝐶)
4	3	eqriv 2733	1 ⊢ (𝐴 ∩ 𝐵) = 𝐶

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ∩ cin 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-in 3957

This theorem is referenced by:  inidm  4226  inass  4227  dfin2  4270  indi  4283  inab  4308  in0  4394  pwin  5573  dfres3  6001  dmres  6029  inixp  37736