Theorem inteq 3930

Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.)

Assertion

Ref	Expression
inteq	⊢ (A = B → ∩A = ∩B)

Proof of Theorem inteq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 2808	. . 3 ⊢ (A = B → (∀y ∈ A x ∈ y ↔ ∀y ∈ B x ∈ y))
2	1	abbidv 2468	. 2 ⊢ (A = B → {x ∣ ∀y ∈ A x ∈ y} = {x ∣ ∀y ∈ B x ∈ y})
3		dfint2 3929	. 2 ⊢ ∩A = {x ∣ ∀y ∈ A x ∈ y}
4		dfint2 3929	. 2 ⊢ ∩B = {x ∣ ∀y ∈ B x ∈ y}
5	2, 3, 4	3eqtr4g 2410	1 ⊢ (A = B → ∩A = ∩B)

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∩cint 3927

This proof depends on 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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2620  df-int 3928

This theorem is used by:  inteqi  3931  inteqd  3932  unissint  3951  uniintsn  3964  rint0  3967  clos1eq1  5875  clos1eq2  5876