Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  spALT Structured version   Visualization version   GIF version

Theorem spALT 41765
Description: sp 2179 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2179 instead. (New usage is discouraged.)
Assertion
Ref Expression
spALT (∀𝑥𝜑𝜑)

Proof of Theorem spALT
StepHypRef Expression
1 ax-1 6 . . 3 (∀𝑥𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜑))
21axc4i 2319 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑))
3 axc10 2386 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
42, 3syl 17 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174  ax-13 2373
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator