MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equs5av Structured version   Visualization version   GIF version

Theorem equs5av 2275
Description: A property related to substitution that replaces the distinctor from equs5 2463 to a disjoint variable condition. Version of equs5a 2460 with a disjoint variable condition, which does not require ax-13 2375. See also sbalex 2240. (Contributed by NM, 2-Feb-2007.) (Revised by GG, 15-Dec-2023.)
Assertion
Ref Expression
equs5av (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem equs5av
StepHypRef Expression
1 nfa1 2149 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12v2 2177 . . . 4 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32spsd 2185 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
43imp 406 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
51, 4exlimi 2215 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781
This theorem is referenced by:  bj-equs45fv  36794
  Copyright terms: Public domain W3C validator