MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpteq2iaOLD Structured version   Visualization version   GIF version

Theorem mpteq2iaOLD 5245
Description: Obsolete version of mpteq2ia 5244 as of 11-Nov-2024. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
mpteq2ia.1 (𝑥𝐴𝐵 = 𝐶)
Assertion
Ref Expression
mpteq2iaOLD (𝑥𝐴𝐵) = (𝑥𝐴𝐶)

Proof of Theorem mpteq2iaOLD
StepHypRef Expression
1 eqid 2726 . . 3 𝐴 = 𝐴
21ax-gen 1789 . 2 𝑥 𝐴 = 𝐴
3 mpteq2ia.1 . . 3 (𝑥𝐴𝐵 = 𝐶)
43rgen 3057 . 2 𝑥𝐴 𝐵 = 𝐶
5 mpteq12f 5229 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
62, 4, 5mp2an 689 1 (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  wcel 2098  wral 3055  cmpt 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-opab 5204  df-mpt 5225
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator