MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpteq12 Structured version   Visualization version   GIF version

Theorem mpteq12 4696
Description: An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)
Assertion
Ref Expression
mpteq12 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem mpteq12
StepHypRef Expression
1 ax-5 1836 . 2 (𝐴 = 𝐶 → ∀𝑥 𝐴 = 𝐶)
2 mpteq12f 4691 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
31, 2sylan 488 1 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wral 2907  cmpt 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-opab 4674  df-mpt 4675
This theorem is referenced by:  mpteq1  4697  mpteqb  6255  fmptcof  6352  mapxpen  8070  prodeq2w  14567  prdsdsval2  16065  prdsdsval3  16066  ablfac2  18409  mdetunilem9  20345  mdetmul  20348  xkocnv  21527  voliun  23229  itgeq1f  23444  itgeq2  23450  iblcnlem  23461  esumeq2  29876  esumcvg  29926  dvtan  33089  bddiblnc  33109
  Copyright terms: Public domain W3C validator