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Mirrors > Home > MPE Home > Th. List > necon2d | Structured version Visualization version GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.) |
Ref | Expression |
---|---|
necon2d.1 | ⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)) |
Ref | Expression |
---|---|
necon2d | ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2d.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝐶 ≠ 𝐷)) | |
2 | df-ne 2939 | . . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
3 | 1, 2 | imbitrdi 251 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)) |
4 | 3 | necon2ad 2953 | 1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ≠ wne 2938 |
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 207 df-ne 2939 |
This theorem is referenced by: map0g 8923 cantnf 9731 hashprg 14431 bcthlem5 25376 deg1ldgn 26147 cxpeq0 26735 lfgrn1cycl 29835 uspgrn2crct 29838 poimirlem17 37624 poimirlem20 37627 poimirlem22 37629 poimirlem27 37634 islshpat 38999 cdleme18b 40275 cdlemh 40800 prjspner1 42613 |
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