Theorem nineq1 3235

Description: Equality law for anti-intersection. (Contributed by SF, 11-Jan-2015.)

Assertion

Ref	Expression
nineq1	⊢ (A = B → (A ⩃ C) = (B ⩃ C))

Proof of Theorem nineq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	nanbi1d 1301	. . 3 ⊢ (A = B → ((x ∈ A ⊼ x ∈ C) ↔ (x ∈ B ⊼ x ∈ C)))
3	2	abbidv 2468	. 2 ⊢ (A = B → {x ∣ (x ∈ A ⊼ x ∈ C)} = {x ∣ (x ∈ B ⊼ x ∈ C)})
4		df-nin 3212	. 2 ⊢ (A ⩃ C) = {x ∣ (x ∈ A ⊼ x ∈ C)}
5		df-nin 3212	. 2 ⊢ (B ⩃ C) = {x ∣ (x ∈ B ⊼ x ∈ C)}
6	3, 4, 5	3eqtr4g 2410	1 ⊢ (A = B → (A ⩃ C) = (B ⩃ C))

Syntax hints:   → wi 4   ⊼ wnan 1287   = wceq 1642   ∈ wcel 1710  {cab 2339   ⩃ cnin 3205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nin 3212

This theorem is referenced by:  nineq2  3236  nineq12  3237  nineq1i  3238  nineq1d  3241  difeq1  3247  ninexg  4098