ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  equs45f GIF version

Theorem equs45f 1756
Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 25-Apr-2008.)
Hypothesis
Ref Expression
equs45f.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
equs45f (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs45f
StepHypRef Expression
1 equs45f.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
21anim2i 337 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑))
32eximi 1562 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
4 equs5a 1748 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
53, 4syl 14 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
6 equs4 1686 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6impbii 125 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  wex 1451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-11 1467  ax-4 1470  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  sb5f  1758
  Copyright terms: Public domain W3C validator