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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 3017 | . . 3 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
3 | 1, 2 | syl6ib 252 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝐶 = 𝐷)) |
4 | 3 | necon2ad 3031 | 1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ≠ wne 3016 |
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 208 df-ne 3017 |
This theorem is referenced by: map0g 8438 cantnf 9145 hashprg 13746 bcthlem5 23860 deg1ldgn 24616 cxpeq0 25188 lfgrn1cycl 27511 uspgrn2crct 27514 poimirlem17 34791 poimirlem20 34794 poimirlem22 34796 poimirlem27 34801 islshpat 36035 cdleme18b 37310 cdlemh 37835 |
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