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Theorem sqxpeqd 4565
Description: Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.)
Hypothesis
Ref Expression
xpeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
sqxpeqd (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵))

Proof of Theorem sqxpeqd
StepHypRef Expression
1 xpeq1d.1 . 2 (𝜑𝐴 = 𝐵)
21, 1xpeq12d 4564 1 (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331   × cxp 4537
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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-opab 3990  df-xp 4545
This theorem is referenced by:  ispsmet  12502  isxms  12630  isms  12632  xmspropd  12656  mspropd  12657
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