Theorem neleq2 2889

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 2611	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
2		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	1, 2	neleq12d 2887	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∉ wnel 2781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-nel 2783

This theorem is referenced by:  noinfep  8440  wrdlndm  13176  isfbas  21443  nbgra0nb  25958  cusgrares  26001  frgrawopreglem4  26574  nbgrnvtx0  40563  nbupgrres  40592  eupth2lem3lem6  41401  frgrwopreglem4  41484