Theorem neleq2 3050

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 2729	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
2		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	1, 2	neleq12d 3048	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∉ wnel 3043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2720  df-clel 2806  df-nel 3044

This theorem is referenced by:  noinfep  9684  isfbas  23746  upgrreslem  29130  umgrreslem  29131  nbgrnvtx0  29165  nbupgrres  29190  eupth2lem3lem6  30056  frgrncvvdeqlem1  30122  frgrwopreglem4a  30133