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Theorem mpteq12dva 3829
Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpteq12dv.1 (φA = 𝐶)
mpteq12dva.2 ((φ x A) → B = 𝐷)
Assertion
Ref Expression
mpteq12dva (φ → (x AB) = (x 𝐶𝐷))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   𝐶(x)   𝐷(x)

Proof of Theorem mpteq12dva
StepHypRef Expression
1 mpteq12dv.1 . . 3 (φA = 𝐶)
21alrimiv 1751 . 2 (φx A = 𝐶)
3 mpteq12dva.2 . . 3 ((φ x A) → B = 𝐷)
43ralrimiva 2386 . 2 (φx A B = 𝐷)
5 mpteq12f 3828 . 2 ((x A = 𝐶 x A B = 𝐷) → (x AB) = (x 𝐶𝐷))
62, 4, 5syl2anc 391 1 (φ → (x AB) = (x 𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  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:  mpteq12dv  3830
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