MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsb1ALT Structured version   Visualization version   GIF version

Theorem equsb1ALT 2601
Description: Alternate version of equsb1 2530. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.p4 (𝜃 ↔ ((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦)))
Assertion
Ref Expression
equsb1ALT 𝜃

Proof of Theorem equsb1ALT
StepHypRef Expression
1 dfsb1.p4 . . 3 (𝜃 ↔ ((𝑥 = 𝑦𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦𝑥 = 𝑦)))
21sb2ALT 2587 . 2 (∀𝑥(𝑥 = 𝑦𝑥 = 𝑦) → 𝜃)
3 id 22 . 2 (𝑥 = 𝑦𝑥 = 𝑦)
42, 3mpg 1799 1 𝜃
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  sbieALT  2613
  Copyright terms: Public domain W3C validator