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Theorem bj-axc14 33167
Description: Alternate proof of axc14 2518 (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 33166 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧 𝑥𝑦))
2 nf5r 2217 . . 3 (Ⅎ𝑧 𝑥𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦))
32a1i 11 . 2 (¬ ∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧 𝑥𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
41, 3syld 47 1 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1628  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857
This theorem is referenced by: (None)
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