New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sb8mo GIF version

Theorem sb8mo 2223
 Description: Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
sb8eu.1 yφ
Assertion
Ref Expression
sb8mo (∃*xφ∃*y[y / x]φ)

Proof of Theorem sb8mo
StepHypRef Expression
1 sb8eu.1 . . . 4 yφ
21sb8e 2093 . . 3 (xφy[y / x]φ)
31sb8eu 2222 . . 3 (∃!xφ∃!y[y / x]φ)
42, 3imbi12i 316 . 2 ((xφ∃!xφ) ↔ (y[y / x]φ∃!y[y / x]φ))
5 df-mo 2209 . 2 (∃*xφ ↔ (xφ∃!xφ))
6 df-mo 2209 . 2 (∃*y[y / x]φ ↔ (y[y / x]φ∃!y[y / x]φ))
74, 5, 63bitr4i 268 1 (∃*xφ∃*y[y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544  [wsb 1648  ∃!weu 2204  ∃*wmo 2205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator