Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ax16 GIF version

Theorem ax16 1785
 Description: Theorem showing that ax-16 1786 is redundant if ax-17 1506 is included in the axiom system. The important part of the proof is provided by aev 1784. See ax16ALT 1831 for an alternate proof that does not require ax-10 1483 or ax-12 1489. This theorem should not be referenced in any proof. Instead, use ax-16 1786 below so that theorems needing ax-16 1786 can be more easily identified. (Contributed by NM, 8-Nov-2006.)
Assertion
Ref Expression
ax16 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax16
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 aev 1784 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
2 ax-17 1506 . . . 4 (𝜑 → ∀𝑧𝜑)
3 sbequ12 1744 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
43biimpcd 158 . . . 4 (𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑥]𝜑))
52, 4alimdh 1443 . . 3 (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧[𝑧 / 𝑥]𝜑))
62hbsb3 1780 . . . 4 ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
7 stdpc7 1743 . . . 4 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑𝜑))
86, 2, 7cbv3h 1721 . . 3 (∀𝑧[𝑧 / 𝑥]𝜑 → ∀𝑥𝜑)
95, 8syl6com 35 . 2 (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑))
101, 9syl 14 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1329  [wsb 1735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736 This theorem is referenced by:  dveeq2  1787  dveeq2or  1788  a16g  1836  exists2  2096
 Copyright terms: Public domain W3C validator