Theorem inteq 1968

Description: Equality law for intersection.

Assertion

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

Proof of Theorem inteq

Step	Hyp	Ref	Expression
1		raleq 1324	. . 3 ⊢ (A = B → (∀y ∈ A x ∈ y ↔ ∀y ∈ B x ∈ y))
2	1	biabdv 1183	. 2 ⊢ (A = B → {x∣∀y ∈ A x ∈ y} = {x∣∀y ∈ B x ∈ y})
3		dfint2 1967	. 2 ⊢ ∩A = {x∣∀y ∈ A x ∈ y}
4		dfint2 1967	. 2 ⊢ ∩B = {x∣∀y ∈ B x ∈ y}
5	2, 3, 4	3eqtr4g 1147	1 ⊢ (A = B → ∩A = ∩B)

Syntax hints:   → wi 2   ∈ wel 803  {cab 1090   = wceq 1091  ∀wral 1201  ∩cint 1965

This theorem is referenced by:  inteqi 1969  inteqd 1970  intex 1986  elreldm 2554  elxp5 2641  fundmen 3333  xpsnen 3339  mapunen 3397  fiint 3445  xpnnen 4927  shintclt 5295  chintclt 5297

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-int 1966

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