Theorem neleq2 3043

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 2736	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
2		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	1, 2	neleq12d 3041	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∉ wnel 3036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-clel 2809  df-nel 3037

This theorem is referenced by:  noinfep  9674  isfbas  23767  upgrreslem  29283  umgrreslem  29284  nbgrnvtx0  29318  nbupgrres  29343  eupth2lem3lem6  30214  frgrncvvdeqlem1  30280  frgrwopreglem4a  30291  clnbgrnvtx0  47841