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Theorem txpeq1 5779
 Description: Equality theorem for tail cross product. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
txpeq1 (A = B → (AC) = (BC))

Proof of Theorem txpeq1
StepHypRef Expression
1 coeq2 4875 . . 3 (A = B → (1st A) = (1st B))
21ineq1d 3456 . 2 (A = B → ((1st A) ∩ (2nd C)) = ((1st B) ∩ (2nd C)))
3 df-txp 5736 . 2 (AC) = ((1st A) ∩ (2nd C))
4 df-txp 5736 . 2 (BC) = ((1st B) ∩ (2nd C))
52, 3, 43eqtr4g 2410 1 (A = B → (AC) = (BC))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∩ cin 3208  1st c1st 4717   ∘ ccom 4721  ◡ccnv 4771  2nd c2nd 4783   ⊗ ctxp 5735 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726  df-txp 5736 This theorem is referenced by:  pprodeq1  5834
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