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

Theorem equsalhw 1838
 Description: Weaker version of equsalh 1961 (requiring distinct variables) without using ax-12 1925. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 28-Dec-2017.)
Hypotheses
Ref Expression
equsalhw.1 (ψxψ)
equsalhw.2 (x = y → (φψ))
Assertion
Ref Expression
equsalhw (x(x = yφ) ↔ ψ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem equsalhw
StepHypRef Expression
1 equsalhw.1 . . 3 (ψxψ)
2119.23h 1802 . 2 (x(x = yψ) ↔ (x x = yψ))
3 equsalhw.2 . . . 4 (x = y → (φψ))
43pm5.74i 236 . . 3 ((x = yφ) ↔ (x = yψ))
54albii 1566 . 2 (x(x = yφ) ↔ x(x = yψ))
6 a9ev 1656 . . 3 x x = y
76a1bi 327 . 2 (ψ ↔ (x x = yψ))
82, 5, 73bitr4i 268 1 (x(x = yφ) ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545 This theorem is referenced by:  dvelimhw  1849
 Copyright terms: Public domain W3C validator