Theorem inteq 3639

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 2549	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦))
2	1	abbidv 2196	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦})
3		dfint2 3638	. 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦}
4		dfint2 3638	. 2 ⊢ ∩ 𝐵 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 𝑥 ∈ 𝑦}
5	2, 3, 4	3eqtr4g 2138	1 ⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵)

Syntax hints:   → wi 4   = wceq 1284  {cab 2067  ∀wral 2348  ∩ cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-int 3637

This theorem is referenced by:  inteqi  3640  inteqd  3641  uniintsnr  3672  rint0  3675  intexr  3925  onintexmid  4315  elreldm  4578  elxp5  4829  1stval2  5802  fundmen  6309  xpsnen  6318  bj-intexr  10699