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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 2992 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
4 | 1, 3 | pm2.21dd 198 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ≠ wne 2987 |
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 210 df-ne 2988 |
This theorem is referenced by: cshwshashlem2 16422 dprdsn 19151 ablsimpgfind 19225 coseq00topi 25095 tglndim0 26423 ncolncol 26440 footne 26517 s3f1 30649 cycpmco2lem7 30824 linds2eq 30995 sgnsub 31912 sgnmulsgn 31917 sgnmulsgp 31918 pconnconn 32591 irrdifflemf 34739 osumcllem11N 37262 dochexmidlem8 38763 remul01 39545 fnchoice 41658 |
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