Theorem ineq1 3100

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 2075	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	anbi1d 438	. . 3 ⊢ (A = B → ((x ∈ A ∧ x ∈ 𝐶) ↔ (x ∈ B ∧ x ∈ 𝐶)))
3		elin 3095	. . 3 ⊢ (x ∈ (A ∩ 𝐶) ↔ (x ∈ A ∧ x ∈ 𝐶))
4		elin 3095	. . 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 2012	1 ⊢ (A = B → (A ∩ 𝐶) = (B ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1224   ∈ wcel 1367   ∩ cin 2885

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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996

This theorem depends on definitions:  df-bi 110  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-v 2529  df-in 2893

This theorem is referenced by:  ineq2  3101  ineq12  3102  ineq1i  3103  ineq1d  3106  dfrab3ss  3184  intprg  3612  inex1g  3857  reseq1  4522  bdinex1g  8354