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Theorem spALT 40826
Description: sp 2184 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2184 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 2343 . 2 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑))
3 axc10 2405 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
42, 3syl 17 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536
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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator