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Theorem mpteq2da 3837
 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 AB) = (x A𝐶))

Proof of Theorem mpteq2da
StepHypRef Expression
1 eqid 2037 . . 3 A = A
21ax-gen 1335 . 2 x A = A
3 mpteq2da.1 . . 3 xφ
4 mpteq2da.2 . . . 4 ((φ x A) → B = 𝐶)
54ex 108 . . 3 (φ → (x AB = 𝐶))
63, 5ralrimi 2384 . 2 (φx A B = 𝐶)
7 mpteq12f 3828 . 2 ((x A = A x A B = 𝐶) → (x AB) = (x A𝐶))
82, 6, 7sylancr 393 1 (φ → (x AB) = (x A𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  ∀wral 2300   ↦ cmpt 3809 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-opab 3810  df-mpt 3811 This theorem is referenced by:  mpteq2dva  3838
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