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Theorem mpteq12da 40380
 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 2199 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐶)
4 mpteq12da.3 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
54ex 403 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐷))
61, 5ralrimi 3139 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐷)
7 mpteq12f 4969 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
83, 6, 7syl2anc 579 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599   = wceq 1601  Ⅎwnf 1827   ∈ wcel 2107  ∀wral 3090   ↦ cmpt 4967 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-opab 4951  df-mpt 4968 This theorem is referenced by:  smflimmpt  41957  smflimsupmpt  41976  smfliminfmpt  41979
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