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Theorem mpteq12dv 3830
Description: An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq12dv.1 (φA = 𝐶)
mpteq12dv.2 (φB = 𝐷)
Assertion
Ref Expression
mpteq12dv (φ → (x AB) = (x 𝐶𝐷))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   𝐶(x)   𝐷(x)

Proof of Theorem mpteq12dv
StepHypRef Expression
1 mpteq12dv.1 . 2 (φA = 𝐶)
2 mpteq12dv.2 . . 3 (φB = 𝐷)
32adantr 261 . 2 ((φ x A) → B = 𝐷)
41, 3mpteq12dva 3829 1 (φ → (x AB) = (x 𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  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:  mpteq12i  3836  offval  5661  offval3  5703
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