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

Theorem equsalhw 2292
Description: Version of equsalh 2419 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
Hypotheses
Ref Expression
equsalhw.1 (𝜓 → ∀𝑥𝜓)
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 (𝜓 → ∀𝑥𝜓)
21nf5i 2146 . 2 𝑥𝜓
3 equsalhw.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3equsalv 2264 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175
This theorem depends on definitions:  df-bi 210  df-ex 1788  df-nf 1792
This theorem is referenced by:  dvelimhw  2346
  Copyright terms: Public domain W3C validator