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Theorem ax12b 2360
Description: A bidirectional version of axc15 2357. (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
ax12b ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax12b
StepHypRef Expression
1 axc15 2357 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
21imp 398 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 sp 2111 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
43com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
54adantl 474 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
62, 5impbid 204 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wal 1505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-10 2079  ax-12 2106  ax-13 2301
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747
This theorem is referenced by: (None)
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