Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  a16gb GIF version

Theorem a16gb 1745
 Description: A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
a16gb (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem a16gb
StepHypRef Expression
1 a16g 1744 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
2 ax-4 1400 . 2 (∀𝑧𝜑𝜑)
31, 2impbid1 130 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator