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Theorem mpteq2daOLD 5240
Description: Obsolete version of mpteq2da 5239 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 2736 . . 3 𝐴 = 𝐴
21ax-gen 1794 . 2 𝑥 𝐴 = 𝐴
3 mpteq2da.1 . . 3 𝑥𝜑
4 mpteq2da.2 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
54ex 412 . . 3 (𝜑 → (𝑥𝐴𝐵 = 𝐶))
63, 5ralrimi 3256 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐶)
7 mpteq12f 5229 . 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 1537   = wceq 1539  wnf 1782  wcel 2107  wral 3060  cmpt 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-opab 5205  df-mpt 5225
This theorem is referenced by: (None)
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