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Theorem mpteq2da 5666
 Description: Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq2da.1 xφ
mpteq2da.2 ((φ x A) → B = C)
Assertion
Ref Expression
mpteq2da (φ → (x A B) = (x A C))

Proof of Theorem mpteq2da
StepHypRef Expression
1 eqid 2353 . . 3 A = A
21ax-gen 1546 . 2 x A = A
3 mpteq2da.1 . . 3 xφ
4 mpteq2da.2 . . . 4 ((φ x A) → B = C)
54ex 423 . . 3 (φ → (x AB = C))
63, 5ralrimi 2695 . 2 (φx A B = C)
7 mpteq12f 5655 . 2 ((x A = A x A B = C) → (x A B) = (x A C))
82, 6, 7sylancr 644 1 (φ → (x A B) = (x A C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = 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:  mpteq2dva  5667
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