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

Theorem ax11w 1993
 Description: Weak version of ax-11 2020 from which we can prove any ax-11 2020 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2020, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
ax11w.1 (𝑦 = 𝑧 → (𝜑𝜓))
Assertion
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 1957 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 194  ∀wal 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921 This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator