Theorem sqxpeqd 4719

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 4718	1 |- ( ph -> ( A X. A ) = ( B X. B ) )

Syntax hints:   -> wi 4   = wceq 1373   X. cxp 4691

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-opab 4122  df-xp 4699

This theorem is referenced by:  prdsval  13220  imasaddfnlemg  13261  intopsn  13314  srg1zr  13864  ispsmet  14910  isxms  15038  isms  15040  xmspropd  15064  mspropd  15065