Theorem ineq1 3358

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 2260	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1d 465	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
3		elin 3347	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
4		elin 3347	. . 3 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
5	2, 3, 4	3bitr4g 223	. 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ 𝑥 ∈ (𝐵 ∩ 𝐶)))
6	5	eqrdv 2194	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ∩ cin 3156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163

This theorem is referenced by:  ineq2  3359  ineq12  3360  ineq1i  3361  ineq1d  3364  dfrab3ss  3442  intprg  3908  inex1g  4170  reseq1  4941  fiintim  7001  uzin2  11169  ressvalsets  12767  elrestr  12949  tgval  12964  inopn  14323  isbasisg  14364  basis1  14367  basis2  14368  ntrfval  14420  tgrest  14489  restco  14494  restsn  14500  restopnb  14501  txrest  14596  metrest  14826  qtopbasss  14841  bdinex1g  15631