Theorem inab 3523

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 2069	. . 3 ⊢ ([y / x](φ ∧ ψ) ↔ ([y / x]φ ∧ [y / x]ψ))
2		df-clab 2340	. . 3 ⊢ (y ∈ {x ∣ (φ ∧ ψ)} ↔ [y / x](φ ∧ ψ))
3		df-clab 2340	. . . 4 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ)
4		df-clab 2340	. . . 4 ⊢ (y ∈ {x ∣ ψ} ↔ [y / x]ψ)
5	3, 4	anbi12i 678	. . 3 ⊢ ((y ∈ {x ∣ φ} ∧ y ∈ {x ∣ ψ}) ↔ ([y / x]φ ∧ [y / x]ψ))
6	1, 2, 5	3bitr4ri 269	. 2 ⊢ ((y ∈ {x ∣ φ} ∧ y ∈ {x ∣ ψ}) ↔ y ∈ {x ∣ (φ ∧ ψ)})
7	6	ineqri 3450	1 ⊢ ({x ∣ φ} ∩ {x ∣ ψ}) = {x ∣ (φ ∧ ψ)}

Syntax hints:   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339   ∩ 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:  inrab  3528  inrab2  3529  dfrab2  3531  dfrab3  3532  evenodddisj  4517  spfinex  4538  pmex  6006  nmembers1lem1  6269