Theorem neleq2 3081

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Assertion

Ref	Expression
neleq2	⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Proof of Theorem neleq2

Step	Hyp	Ref	Expression
1		eqidd 2781	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
2		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	1, 2	neleq12d 3079	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1508   ∉ wnel 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-ext 2752

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-cleq 2773  df-clel 2848  df-nel 3076

This theorem is referenced by:  noinfep  8923  wrdlndmOLD  13697  isfbas  22156  upgrreslem  26804  umgrreslem  26805  nbgrnvtx0  26839  nbupgrres  26864  eupth2lem3lem6  27778  frgrncvvdeqlem1  27848  frgrwopreglem4a  27859