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Theorem mpteq12da 41699
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 2214 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐶)
4 mpteq12da.3 . . 3 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
51, 4ralrimia 41584 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐷)
6 mpteq12f 5122 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
73, 5, 6syl2anc 587 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wnf 1785  wcel 2115  wral 3126  cmpt 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ral 3131  df-opab 5102  df-mpt 5120
This theorem is referenced by:  smflimmpt  43264  smflimsupmpt  43283  smfliminfmpt  43286
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