Theorem ineq1 3108

Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)

Assertion

Ref	Expression
ineq1	⊢ (A = B → (A ∩ 𝐶) = (B ∩ 𝐶))

Proof of Theorem ineq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2083	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	anbi1d 441	. . 3 ⊢ (A = B → ((x ∈ A ∧ x ∈ 𝐶) ↔ (x ∈ B ∧ x ∈ 𝐶)))
3		elin 3103	. . 3 ⊢ (x ∈ (A ∩ 𝐶) ↔ (x ∈ A ∧ x ∈ 𝐶))
4		elin 3103	. . 3 ⊢ (x ∈ (B ∩ 𝐶) ↔ (x ∈ B ∧ x ∈ 𝐶))
5	2, 3, 4	3bitr4g 212	. 2 ⊢ (A = B → (x ∈ (A ∩ 𝐶) ↔ x ∈ (B ∩ 𝐶)))
6	5	eqrdv 2020	1 ⊢ (A = B → (A ∩ 𝐶) = (B ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ∩ cin 2893

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901

This theorem is referenced by:  ineq2  3109  ineq12  3110  ineq1i  3111  ineq1d  3114  dfrab3ss  3192  intprg  3622  inex1g  3867  reseq1  4533  bdinex1g  7124