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Theorem axextbdist 34414
Description: axextb 2711 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
axextbdist ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))

Proof of Theorem axextbdist
StepHypRef Expression
1 axc9 2381 . . . 4 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
21imp 408 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
3 nfnae 2433 . . . . 5 𝑧 ¬ ∀𝑧 𝑧 = 𝑥
4 nfnae 2433 . . . . 5 𝑧 ¬ ∀𝑧 𝑧 = 𝑦
53, 4nfan 1903 . . . 4 𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
6 elequ2 2122 . . . . 5 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
76a1i 11 . . . 4 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦)))
85, 7alimd 2206 . . 3 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧 𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦)))
92, 8syld 47 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦)))
10 axextdist 34413 . 2 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
119, 10impbid 211 1 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-clel 2815  df-nfc 2890
This theorem is referenced by: (None)
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