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Mirrors > Home > MPE Home > Th. List > pm2.61iine | Structured version Visualization version GIF version |
Description: Equality version of pm2.61ii 176. (Contributed by Scott Fenton, 13-Jun-2013.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) |
Ref | Expression |
---|---|
pm2.61iine.1 | ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → 𝜑) |
pm2.61iine.2 | ⊢ (𝐴 = 𝐶 → 𝜑) |
pm2.61iine.3 | ⊢ (𝐵 = 𝐷 → 𝜑) |
Ref | Expression |
---|---|
pm2.61iine | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.61iine.2 | . 2 ⊢ (𝐴 = 𝐶 → 𝜑) | |
2 | pm2.61iine.3 | . . . 4 ⊢ (𝐵 = 𝐷 → 𝜑) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 = 𝐷) → 𝜑) |
4 | pm2.61iine.1 | . . 3 ⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → 𝜑) | |
5 | 3, 4 | pm2.61dane 2869 | . 2 ⊢ (𝐴 ≠ 𝐶 → 𝜑) |
6 | 1, 5 | pm2.61ine 2865 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ≠ wne 2780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ne 2782 |
This theorem is referenced by: fntpb 6378 elfiun 8219 dedekind 10079 mdsymi 28654 |
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