Theorem neleq2 3046

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 2741	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
2		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
3	1, 2	neleq12d 3044	1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∉ wnel 3039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2732  df-clel 2815  df-nel 3040

This theorem is referenced by:  noinfep  9579  isfbas  23819  upgrreslem  29398  umgrreslem  29399  nbgrnvtx0  29433  nbupgrres  29458  eupth2lem3lem6  30328  frgrncvvdeqlem1  30394  frgrwopreglem4a  30405  clnbgrnvtx0  48325