Theorem neleq1 3038

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Assertion

Ref	Expression
neleq1	⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))

Proof of Theorem neleq1

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2		eqidd 2732	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
3	1, 2	neleq12d 3037	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∉ wnel 3032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-clel 2806  df-nel 3033

This theorem is referenced by:  ruALT  9492  ssnn0fi  13889  cnpart  15144  sqrmo  15155  resqrtcl  15157  resqrtthlem  15158  sqrtneg  15171  sqreu  15265  sqrtthlem  15267  eqsqrtd  15272  ge2nprmge4  16609  prmgaplem7  16966  mgmnsgrpex  18836  sgrpnmndex  18837  iccpnfcnv  24867  griedg0prc  29240  nbgrssovtx  29337  rgrusgrprc  29566  rusgrprc  29567  rgrprcx  29569  frgrwopreglem4a  30285  xrge0iifcnv  33941  0nn0m1nnn0  35145  fpprel  47758  gpg5nbgrvtx03star  48110  gpg5nbgr3star  48111  grlimedgnedg  48161  oddinmgm  48205