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Mirrors > Home > ILE Home > Th. List > necon1abiidc | GIF version |
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.) |
Ref | Expression |
---|---|
necon1abiidc.1 | ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵)) |
Ref | Expression |
---|---|
necon1abiidc | ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2206 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon1abiidc.1 | . . 3 ⊢ (DECID 𝜑 → (¬ 𝜑 ↔ 𝐴 = 𝐵)) | |
3 | 2 | con1biidc 771 | . 2 ⊢ (DECID 𝜑 → (¬ 𝐴 = 𝐵 ↔ 𝜑)) |
4 | 1, 3 | syl5bb 181 | 1 ⊢ (DECID 𝜑 → (𝐴 ≠ 𝐵 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 DECID wdc 742 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-ne 2206 |
This theorem is referenced by: necon2abiidc 2269 |
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