Theorem ineqri 4142

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 3899	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		ineqri.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)
3	1, 2	bitri 276	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝐶)
4	3	eqriv 2736	1 ⊢ (𝐴 ∩ 𝐵) = 𝐶

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-in 3890

This theorem is referenced by:  inidm  4156  inass  4157  dfin2  4200  indi  4213  inab  4238  in0  4324  pwin  5510  dfres3  5937  dmres  5965  inixp  38104