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Mirrors > Home > ILE Home > Th. List > neeq12d | GIF version |
Description: Deduction for inequality. (Contributed by NM, 24-Jul-2012.) |
Ref | Expression |
---|---|
neeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
neeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
neeq12d | ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | neeq1d 2365 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
3 | neeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | neeq2d 2366 | . 2 ⊢ (𝜑 → (𝐵 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)) |
5 | 2, 4 | bitrd 188 | 1 ⊢ (𝜑 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-ne 2348 |
This theorem is referenced by: 3netr3d 2379 3netr4d 2380 exmidapne 7258 ennnfonelemim 12424 ctinfom 12428 isnzr 13323 apdiff 14766 |
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