Theorem sqxpeqd 4751

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	|- ( ph -> A = B )

Assertion

Ref	Expression
sqxpeqd	|- ( ph -> ( A X. A ) = ( B X. B ) )

Proof of Theorem sqxpeqd

Step	Hyp	Ref	Expression
1		xpeq1d.1	. 2 |- ( ph -> A = B )
2	1, 1	xpeq12d 4750	1 |- ( ph -> ( A X. A ) = ( B X. B ) )

Syntax hints:   -> wi 4   = wceq 1397   X. cxp 4723

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-opab 4151  df-xp 4731

This theorem is referenced by:  prdsval  13355  imasaddfnlemg  13396  intopsn  13449  srg1zr  13999  ispsmet  15046  isxms  15174  isms  15176  xmspropd  15200  mspropd  15201