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Mirrors > Home > MPE Home > Th. List > pm2.21ddne | Structured version Visualization version GIF version |
Description: A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
pm2.21ddne.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
pm2.21ddne.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
pm2.21ddne | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.21ddne.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | pm2.21ddne.2 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
3 | 2 | neneqd 2828 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
4 | 1, 3 | pm2.21dd 186 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ≠ wne 2823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 197 df-ne 2824 |
This theorem is referenced by: cshwshashlem2 15850 dprdsn 18481 coseq00topi 24299 tglndim0 25569 ncolncol 25586 footne 25660 sgnsub 30734 sgnmulsgn 30739 sgnmulsgp 30740 pconnconn 31339 osumcllem11N 35570 dochexmidlem8 37073 fnchoice 39502 |
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