ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  equs5e GIF version

Theorem equs5e 1795
Description: A property related to substitution that unlike equs5 1829 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
equs5e (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5e
StepHypRef Expression
1 19.8a 1590 . . . . 5 (𝜑 → ∃𝑦𝜑)
2 hbe1 1495 . . . . 5 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
31, 2syl 14 . . . 4 (𝜑 → ∀𝑦𝑦𝜑)
43anim2i 342 . . 3 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝑦𝜑))
54eximi 1600 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝑦𝜑))
6 equs5a 1794 . 2 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
75, 6syl 14 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-11 1506  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  ax11e  1796  sb4e  1805
  Copyright terms: Public domain W3C validator