MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc16gALT Structured version   Visualization version   GIF version

Theorem axc16gALT 2527
Description: Alternate proof of axc16g 2254 that uses df-sb 2063 and requires ax-10 2138, ax-11 2153, ax-13 2385. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc16gALT (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16gALT
StepHypRef Expression
1 aev 2055 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2 axc16ALT 2526 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3 biidd 263 . . . 4 (∀𝑧 𝑧 = 𝑥 → (𝜑𝜑))
43dral1 2458 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
54biimprd 249 . 2 (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑))
61, 2, 5sylsyld 61 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator