Theorem inab 4200

Description: Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
inab	⊢ ({𝑥 ∣ 𝜑} ∩ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ 𝜓)}

Proof of Theorem inab

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sban 2088	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
2		df-clab 2715	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ 𝜓)} ↔ [𝑦 / 𝑥](𝜑 ∧ 𝜓))
3		df-clab 2715	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4		df-clab 2715	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
5	3, 4	anbi12i 630	. . 3 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
6	1, 2, 5	3bitr4ri 307	. 2 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∧ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 ∧ 𝜓)})
7	6	ineqri 4105	1 ⊢ ({𝑥 ∣ 𝜑} ∩ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ 𝜓)}

Syntax hints:   ∧ wa 399   = wceq 1543  [wsb 2072   ∈ wcel 2112  {cab 2714   ∩ cin 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-in 3860

This theorem is referenced by:  inrab  4207  inrab2  4208  dfrab3  4210  orduniss2  7590  ssenen  8798  hashf1lem2  13987  symgsubmefmnd  18744  ballotlem2  32121  fmla0disjsuc  33027  dfiota3  33911  bj-inrab  34801  ptrest  35462  diophin  40238