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Theorem mpteq1 5658
 Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq1 (A = B → (x A C) = (x B C))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   C(x)

Proof of Theorem mpteq1
StepHypRef Expression
1 eqidd 2354 . . 3 (x AC = C)
21rgen 2679 . 2 x A C = C
3 mpteq12 5657 . 2 ((A = B x A C = C) → (x A C) = (x B C))
42, 3mpan2 652 1 (A = B → (x A C) = (x B C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ↦ cmpt 5651 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2619  df-opab 4623  df-mpt 5652 This theorem is referenced by:  mpt2mpt  5709  elscan  6330
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