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Theorem mpteq2daOLD 5208
Description: Obsolete version of mpteq2da 5207 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 2733 . . 3 𝐴 = 𝐴
21ax-gen 1798 . 2 𝑥 𝐴 = 𝐴
3 mpteq2da.1 . . 3 𝑥𝜑
4 mpteq2da.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
54ex 414 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
63, 5ralrimi 3239 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
7 mpteq12f 5197 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
82, 6, 7sylancr 588 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wnf 1786  wcel 2107  wral 3061  cmpt 5192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-opab 5172  df-mpt 5193
This theorem is referenced by: (None)
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