MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax11w Structured version   Visualization version   GIF version

Theorem ax11w 1954
Description: Weak version of ax-11 1970 from which we can prove any ax-11 1970 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 1970, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)
Ref Expression
ax11w.1 (𝑦 = 𝑧 → (𝜑𝜓))
Ref Expression
ax11w (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Distinct variable groups:   𝑦,𝑧   𝑥,𝑦   𝜑,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑧)

Proof of Theorem ax11w
StepHypRef Expression
1 ax11w.1 . 2 (𝑦 = 𝑧 → (𝜑𝜓))
21alcomiw 1919 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wal 1466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator