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Mirrors > Home > ILE Home > Th. List > xpeq1d | GIF version |
Description: Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Ref | Expression |
---|---|
xpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
xpeq1d | ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | xpeq1 4636 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 × cxp 4620 |
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-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-opab 4062 df-xp 4628 |
This theorem is referenced by: xpssres 4937 ixpsnf1o 6729 xpfi 6922 hashxp 10777 |
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