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Theorem mpteq12da 39965
Description: An equality inference for the maps to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
mpteq12da.1 𝑥𝜑
mpteq12da.2 (𝜑𝐴 = 𝐶)
mpteq12da.3 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12da (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Proof of Theorem mpteq12da
StepHypRef Expression
1 mpteq12da.1 . . 3 𝑥𝜑
2 mpteq12da.2 . . 3 (𝜑𝐴 = 𝐶)
31, 2alrimi 2238 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐶)
4 mpteq12da.3 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
54ex 397 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐷))
61, 5ralrimi 3106 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐷)
7 mpteq12f 4866 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
83, 6, 7syl2anc 573 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629   = wceq 1631  wnf 1856  wcel 2145  wral 3061  cmpt 4864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-opab 4848  df-mpt 4865
This theorem is referenced by:  smflimmpt  41531  smflimsupmpt  41550  smfliminfmpt  41553
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