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Theorem ax13ALT 2362
 Description: Alternate proof of ax13 2305 from FOL, sp 2112, and axc9 2313. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax13ALT 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Proof of Theorem ax13ALT
StepHypRef Expression
1 sp 2112 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
21con3i 152 . . 3 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3 sp 2112 . . . 4 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑧)
43con3i 152 . . 3 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑧)
5 axc9 2313 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
62, 4, 5syl2im 40 . 2 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)))
7 ax13b 1990 . 2 ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))))
86, 7mpbir 223 1 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1506 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-12 2107  ax-13 2302 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748 This theorem is referenced by: (None)
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