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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 2822 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶) | |
2 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
3 | 1, 2 | neleq12d 3127 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∉ wnel 3123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2814 df-clel 2893 df-nel 3124 |
This theorem is referenced by: noinfep 9112 wrdlndmOLD 13870 isfbas 22367 upgrreslem 27014 umgrreslem 27015 nbgrnvtx0 27049 nbupgrres 27074 eupth2lem3lem6 27940 frgrncvvdeqlem1 28006 frgrwopreglem4a 28017 |
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