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Mirrors > Home > MPE Home > Th. List > 3netr3g | Structured version Visualization version GIF version |
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) |
Ref | Expression |
---|---|
3netr3g.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
3netr3g.2 | ⊢ 𝐴 = 𝐶 |
3netr3g.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3netr3g | ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3netr3g.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | 3netr3g.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3netr3g.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | neeq12i 3001 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) |
5 | 1, 4 | sylib 217 | 1 ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ≠ wne 2934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-cleq 2718 df-ne 2935 |
This theorem is referenced by: (None) |
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