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Theorem bj-axc14 32032
 Description: Alternate proof of axc14 2360 (even when inlining the above results, this gives a shorter proof). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc14 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))

Proof of Theorem bj-axc14
StepHypRef Expression
1 bj-axc14nf 32031 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
2 nf5r 2052 . . 3 (Ⅎ𝑧 𝑥𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦))
32a1i 11 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧 𝑥𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
41, 3syld 46 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606 This theorem is referenced by: (None)
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