Theorem inteq 4914

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 3319	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦))
2	1	abbidv 2828	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦})
3		dfint2 4913	. 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
4		dfint2 4913	. 2 ⊢ ∩ 𝐵 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦}
5	2, 3, 4	3eqtr4g 2822	1 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  {cab 2740  ∀wral 3078  ∩ cint 4911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-ral 3079  df-rex 3089  df-int 4912

This theorem is used by:  inteqi  4915  inteqd  4916  unissint  4936  uniintsn  4949  rint0  4952  intex  5313  intnex  5314  elreldm  5924  elxp5  7918  1stval2  8001  oev2  8506  fundmen  9026  xpsnen  9047  fiint  9284  elfir  9373  inelfi  9376  fiin  9380  cardmin2  9992  isfin2-2  10309  incexclem  15897  mreintcl  17653  ismred2  17661  fiinopn  23069  cmpfii  23577  ptbasfi  23749  fbssint  24006  shintcl  31693  chintcl  31695  zarcmplem  34280  inelpisys  34553  rankeq1o  36671  bj-0int  37771  bj-ismoored  37777  bj-snmoore  37783  bj-prmoore  37785  neificl  38432  heibor1lem  38488  elrfi  43453  elrfirn  43454