Theorem unab 3522

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

Assertion

Ref	Expression
unab	⊢ ({x ∣ φ} ∪ {x ∣ ψ}) = {x ∣ (φ ∨ ψ)}

Proof of Theorem unab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbor 2066	. . 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	orbi12i 507	. . 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	uneqri 3407	1 ⊢ ({x ∣ φ} ∪ {x ∣ ψ}) = {x ∣ (φ ∨ ψ)}

Syntax hints:   ∨ wo 357   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339   ∪ cun 3208

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-un 3215

This theorem is referenced by:  unrab  3527  rabun2  3535  dfif6  3666  nnc0suc  4413  nncaddccl  4420  preaddccan2lem1  4455  ltfintrilem1  4466  nnadjoin  4521  tfinnn  4535  phiun  4615  unopab  4639  clos1basesuc  5883  leconnnc  6219  addccan2nclem2  6265  nchoicelem16  6305