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Theorem mpteq2daOLD 5247
Description: Obsolete version of mpteq2da 5246 as of 11-Nov-2024. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
mpteq2da.1 𝑥𝜑
mpteq2da.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
mpteq2daOLD (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))

Proof of Theorem mpteq2daOLD
StepHypRef Expression
1 eqid 2735 . . 3 𝐴 = 𝐴
21ax-gen 1792 . 2 𝑥 𝐴 = 𝐴
3 mpteq2da.1 . . 3 𝑥𝜑
4 mpteq2da.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
54ex 412 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
63, 5ralrimi 3255 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
7 mpteq12f 5236 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
82, 6, 7sylancr 587 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wnf 1780  wcel 2106  wral 3059  cmpt 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-opab 5211  df-mpt 5232
This theorem is referenced by: (None)
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