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

Theorem alcomiw 2043
Description: Weak version of alcom 2155. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Dec-2023.)
Hypothesis
Ref Expression
alcomiw.1 (𝑦 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
alcomiw (∀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 alcomiw
StepHypRef Expression
1 alcomiw.1 . . . . 5 (𝑦 = 𝑧 → (𝜑𝜓))
21cbvalvw 2036 . . . 4 (∀𝑦𝜑 ↔ ∀𝑧𝜓)
32biimpi 218 . . 3 (∀𝑦𝜑 → ∀𝑧𝜓)
43alimi 1805 . 2 (∀𝑥𝑦𝜑 → ∀𝑥𝑧𝜓)
5 ax-5 1904 . 2 (∀𝑥𝑧𝜓 → ∀𝑦𝑥𝑧𝜓)
61biimprd 250 . . . . 5 (𝑦 = 𝑧 → (𝜓𝜑))
76equcoms 2020 . . . 4 (𝑧 = 𝑦 → (𝜓𝜑))
87spimvw 1995 . . 3 (∀𝑧𝜓𝜑)
982alimi 1806 . 2 (∀𝑦𝑥𝑧𝜓 → ∀𝑦𝑥𝜑)
104, 5, 93syl 18 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  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
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774
This theorem is referenced by:  hbalw  2049  ax11w  2127  bj-ssblem2  33976
  Copyright terms: Public domain W3C validator