Theorem neleq1 3043

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 2738	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
3	1, 2	neleq12d 3042	1 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∉ wnel 3037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-clel 2812  df-nel 3038

This theorem is referenced by:  ruALT  9523  ssnn0fi  13920  cnpart  15175  sqrmo  15186  resqrtcl  15188  resqrtthlem  15189  sqrtneg  15202  sqreu  15296  sqrtthlem  15298  eqsqrtd  15303  ge2nprmge4  16640  prmgaplem7  16997  mgmnsgrpex  18868  sgrpnmndex  18869  iccpnfcnv  24910  griedg0prc  29349  nbgrssovtx  29446  rgrusgrprc  29675  rusgrprc  29676  rgrprcx  29678  frgrwopreglem4a  30397  xrge0iifcnv  34110  0nn0m1nnn0  35326  fpprel  48082  gpg5nbgrvtx03star  48434  gpg5nbgr3star  48435  grlimedgnedg  48485  oddinmgm  48529