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Theorem alcomiw 2140
Description: Weak version of alcom 2202. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
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))
21biimpd 221 . . . 4 (𝑦 = 𝑧 → (𝜑𝜓))
32cbvalivw 2106 . . 3 (∀𝑦𝜑 → ∀𝑧𝜓)
43alimi 1907 . 2 (∀𝑥𝑦𝜑 → ∀𝑥𝑧𝜓)
5 ax-5 2006 . 2 (∀𝑥𝑧𝜓 → ∀𝑦𝑥𝑧𝜓)
61biimprd 240 . . . . 5 (𝑦 = 𝑧 → (𝜓𝜑))
76equcoms 2119 . . . 4 (𝑧 = 𝑦 → (𝜓𝜑))
87spimvw 2099 . . 3 (∀𝑧𝜓𝜑)
982alimi 1908 . 2 (∀𝑦𝑥𝑧𝜓 → ∀𝑦𝑥𝜑)
104, 5, 93syl 18 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876
This theorem is referenced by:  hbalw  2146  ax11w  2174  bj-ssblem2  33129
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