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Theorem mpteq2ia 5659
 Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpteq2ia.1 (x AB = C)
Assertion
Ref Expression
mpteq2ia (x A B) = (x A C)

Proof of Theorem mpteq2ia
StepHypRef Expression
1 eqid 2353 . . 3 A = A
21ax-gen 1546 . 2 x A = A
3 mpteq2ia.1 . . 3 (x AB = C)
43rgen 2679 . 2 x A B = C
5 mpteq12f 5655 . 2 ((x A = A x A B = C) → (x A B) = (x A C))
62, 4, 5mp2an 653 1 (x A B) = (x A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ↦ cmpt 5651 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  mpteq2i  5660
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