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Theorem ax13b 2134
 Description: An equivalence between two ways of expressing ax-13 2377. See the comment for ax-13 2377. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)
Assertion
Ref Expression
ax13b ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑))))

Proof of Theorem ax13b
StepHypRef Expression
1 ax-1 6 . . 3 ((𝑦 = 𝑧𝜑) → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑)))
2 equeuclr 2122 . . . . . 6 (𝑦 = 𝑧 → (𝑥 = 𝑧𝑥 = 𝑦))
32con3rr3 153 . . . . 5 𝑥 = 𝑦 → (𝑦 = 𝑧 → ¬ 𝑥 = 𝑧))
43imim1d 82 . . . 4 𝑥 = 𝑦 → ((¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑)) → (𝑦 = 𝑧 → (𝑦 = 𝑧𝜑))))
5 pm2.43 56 . . . 4 ((𝑦 = 𝑧 → (𝑦 = 𝑧𝜑)) → (𝑦 = 𝑧𝜑))
64, 5syl6 35 . . 3 𝑥 = 𝑦 → ((¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑)) → (𝑦 = 𝑧𝜑)))
71, 6impbid2 218 . 2 𝑥 = 𝑦 → ((𝑦 = 𝑧𝜑) ↔ (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑))))
87pm5.74i 263 1 ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107 This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876 This theorem is referenced by:  ax13  2380  ax13ALT  2432  ax13fromc9  34927
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