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

Theorem equs5a 2336
Description: A property related to substitution that unlike equs5 2339 does not require a distinctor antecedent. See equs5aALT 2165 for an alternate proof using ax-12 2034 but not ax13 2237. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
equs5a (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs5a
StepHypRef Expression
1 nfa1 2015 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
2 ax12 2292 . . 3 (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32imp 444 . 2 ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
41, 3exlimi 2073 1 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701
This theorem is referenced by:  equs45f  2338  sb4a  2345  bj-equs45fv  31940
  Copyright terms: Public domain W3C validator