|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > neleq2 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) | 
| Ref | Expression | 
|---|---|
| neleq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqidd 2738 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
| 2 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
| 3 | 1, 2 | neleq12d 3051 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∉ wnel 3046 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-nel 3047 | 
| This theorem is referenced by: noinfep 9700 isfbas 23837 upgrreslem 29321 umgrreslem 29322 nbgrnvtx0 29356 nbupgrres 29381 eupth2lem3lem6 30252 frgrncvvdeqlem1 30318 frgrwopreglem4a 30329 clnbgrnvtx0 47814 | 
| Copyright terms: Public domain | W3C validator |