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Theorem mpteq2ia 3834
 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 AB) = (x A𝐶)

Proof of Theorem mpteq2ia
StepHypRef Expression
1 eqid 2037 . . 3 A = A
21ax-gen 1335 . 2 x A = A
3 mpteq2ia.1 . . 3 (x AB = 𝐶)
43rgen 2368 . 2 x A B = 𝐶
5 mpteq12f 3828 . 2 ((x A = A x A B = 𝐶) → (x AB) = (x A𝐶))
62, 4, 5mp2an 402 1 (x AB) = (x A𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   = wceq 1242   ∈ 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:  mpteq2i  3835  feqresmpt  5170  fmptap  5296  offres  5704  cnrecnv  9118
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