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Theorem ax12b 2448
Description: A bidirectional version of axc15 2446. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
ax12b ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax12b
StepHypRef Expression
1 axc15 2446 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
21imp 410 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 sp 2184 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
43com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
54adantl 485 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
62, 5impbid 215 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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