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Theorem ax11b 1754
Description: A bidirectional version of ax-11o 1751. (Contributed by NM, 30-Jun-2006.)
Assertion
Ref Expression
ax11b ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem ax11b
StepHypRef Expression
1 ax11o 1750 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
21imp 122 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 ax-4 1445 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
43com12 30 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
54adantl 271 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
62, 5impbid 127 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by: (None)
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