Theorem ineq1 3125

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 2098	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	anbi1d 438	. . 3 ⊢ (A = B → ((x ∈ A ∧ x ∈ 𝐶) ↔ (x ∈ B ∧ x ∈ 𝐶)))
3		elin 3120	. . 3 ⊢ (x ∈ (A ∩ 𝐶) ↔ (x ∈ A ∧ x ∈ 𝐶))
4		elin 3120	. . 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 2035	1 ⊢ (A = B → (A ∩ 𝐶) = (B ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ∩ cin 2910

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918

This theorem is referenced by:  ineq2  3126  ineq12  3127  ineq1i  3128  ineq1d  3131  dfrab3ss  3209  intprg  3639  inex1g  3884  reseq1  4549  bdinex1g  9356