Theorem neleq2 3069

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 2762	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
2		id 23	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	1, 2	neleq12d 3067	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∉ wnel 3062

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2753  df-clel 2836  df-nel 3063

This theorem is used by:  noinfep  9645  isfbas  24128  upgrreslem  29867  umgrreslem  29868  nbgrnvtx0  29902  nbupgrres  29927  eupth2lem3lem6  30816  frgrncvvdeqlem1  30882  frgrwopreglem4a  30893  clnbgrnvtx0  48869