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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 4607 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 × cxp 4586 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-opab 4028 df-xp 4594 |
This theorem is referenced by: sqxpeqd 4614 opeliunxp 4643 mpomptsx 6147 dmmpossx 6149 fmpox 6150 disjxp1 6185 erssxp 6505 cc2lem 7188 cc2 7189 fsum2dlemstep 11342 fisumcom2 11346 fprod2dlemstep 11530 fprodcom2fi 11534 txbas 12728 |
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