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Mirrors > Home > ILE Home > Th. List > pm2.21ddne | 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 2368 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
4 | 1, 3 | pm2.21dd 620 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-in2 615 |
This theorem depends on definitions: df-bi 117 df-ne 2348 |
This theorem is referenced by: npnflt 9813 nmnfgt 9816 xlt2add 9878 xrbdtri 11279 divalglemex 11921 divalg 11923 znege1 12172 ennnfonelemex 12409 |
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