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Mirrors > Home > ILE Home > Th. List > necon3ai | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.) |
Ref | Expression |
---|---|
necon3ai.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
necon3ai | ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2309 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon3ai.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | con3i 621 | . 2 ⊢ (¬ 𝐴 = 𝐵 → ¬ 𝜑) |
4 | 1, 3 | sylbi 120 | 1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ≠ wne 2308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-in1 603 ax-in2 604 |
This theorem depends on definitions: df-bi 116 df-ne 2309 |
This theorem is referenced by: disjsn2 3586 0nelxp 4567 fvunsng 5614 map0b 6581 difinfsnlem 6984 hashprg 10554 gcd1 11675 gcdzeq 11710 phimullem 11901 |
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