Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  equs5a Structured version   Visualization version   GIF version

Theorem equs5a 2469
 Description: A property related to substitution that unlike equs5 2472 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2379. This proof uses ax12 2434, see equs5aALT 2373 for an alternative one using ax-12 2175 but not ax13 2382. Usage of the weaker equs5av 2276 is preferred, which uses ax12v2 2177, but not ax-13 2379. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
Assertion
Ref Expression
equs5a (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs5a
StepHypRef Expression
1 nfa1 2152 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12 2434 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32imp 410 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
41, 3exlimi 2215 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  equs45f  2471  sb4aALT  2574
 Copyright terms: Public domain W3C validator