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Mirrors > Home > ILE Home > Th. List > necon3bid | GIF version |
Description: Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon3bid.1 | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Ref | Expression |
---|---|
necon3bid | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2341 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon3bid.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) | |
3 | 2 | necon3bbid 2380 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 ↔ 𝐶 ≠ 𝐷)) |
4 | 1, 3 | syl5bb 191 | 1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1348 ≠ wne 2340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 |
This theorem depends on definitions: df-bi 116 df-ne 2341 |
This theorem is referenced by: nebidc 2420 addneintrd 8107 addneintr2d 8108 negne0bd 8223 negned 8227 subne0d 8239 subne0ad 8241 subneintrd 8274 subneintr2d 8276 qapne 9598 xrlttri3 9754 xaddass2 9827 sqne0 10541 fihashneq0 10729 hashnncl 10730 cjne0 10872 absne0d 11151 sqrt2irraplemnn 12133 metn0 13172 lgsabs1 13734 |
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