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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 2737 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
2 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
3 | 1, 2 | neleq12d 3040 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∉ wnel 3036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2728 df-clel 2809 df-nel 3037 |
This theorem is referenced by: noinfep 9253 isfbas 22680 upgrreslem 27346 umgrreslem 27347 nbgrnvtx0 27381 nbupgrres 27406 eupth2lem3lem6 28270 frgrncvvdeqlem1 28336 frgrwopreglem4a 28347 |
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