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Mirrors > Home > ILE Home > Th. List > xpeq12d | GIF version |
Description: Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.) |
Ref | Expression |
---|---|
xpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
xpeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
xpeq12d | ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | xpeq12d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | xpeq12 4553 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 × cxp 4532 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-opab 3985 df-xp 4540 |
This theorem is referenced by: sqxpeqd 4560 opeliunxp 4589 mpomptsx 6088 dmmpossx 6090 fmpox 6091 disjxp1 6126 erssxp 6445 fsum2dlemstep 11196 fisumcom2 11200 txbas 12416 |
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