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Theorem mpteq2da 3846
 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 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
mpteq2da (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Proof of Theorem mpteq2da
StepHypRef Expression
1 eqid 2040 . . 3 𝐴 = 𝐴
21ax-gen 1338 . 2 𝑥 𝐴 = 𝐴
3 mpteq2da.1 . . 3 𝑥𝜑
4 mpteq2da.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
54ex 108 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
63, 5ralrimi 2390 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
7 mpteq12f 3837 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
82, 6, 7sylancr 393 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306   ↦ cmpt 3818 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820 This theorem is referenced by:  mpteq2dva  3847
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