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Theorem mpteq12daOLD 45249
Description: Obsolete version of mpteq12da 5227 as of 11-Nov-2024. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
mpteq12daOLD.1 𝑥𝜑
mpteq12daOLD.2 (𝜑𝐴 = 𝐶)
mpteq12daOLD.3 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12daOLD (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Proof of Theorem mpteq12daOLD
StepHypRef Expression
1 mpteq12daOLD.1 . . 3 𝑥𝜑
2 mpteq12daOLD.2 . . 3 (𝜑𝐴 = 𝐶)
31, 2alrimi 2213 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐶)
4 mpteq12daOLD.3 . . 3 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
51, 4ralrimia 3258 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐷)
6 mpteq12f 5230 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
73, 5, 6syl2anc 584 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wnf 1783  wcel 2108  wral 3061  cmpt 5225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-opab 5206  df-mpt 5226
This theorem is referenced by: (None)
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