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  9514  ssnn0fi  13938  cnpart  15193  sqrmo  15204  resqrtcl  15206  resqrtthlem  15207  sqrtneg  15220  sqreu  15314  sqrtthlem  15316  eqsqrtd  15321  ge2nprmge4  16662  prmgaplem7  17019  mgmnsgrpex  18893  sgrpnmndex  18894  iccpnfcnv  24921  griedg0prc  29347  nbgrssovtx  29444  rgrusgrprc  29673  rusgrprc  29674  rgrprcx  29676  frgrwopreglem4a  30395  xrge0iifcnv  34093  0nn0m1nnn0  35311  ppivalnnnprm  48103  fpprel  48216  gpg5nbgrvtx03star  48568  gpg5nbgr3star  48569  grlimedgnedg  48619  oddinmgm  48663