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Theorem wl-cbvmotv 34917
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by Wolf Lammen, 5-Mar-2023.)
Assertion
Ref Expression
wl-cbvmotv (∃*𝑥⊤ → ∃*𝑦⊤)
Distinct variable group:   𝑥,𝑦

Proof of Theorem wl-cbvmotv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax7v2 2018 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
21imim2d 57 . . . 4 (𝑥 = 𝑦 → ((⊤ → 𝑥 = 𝑧) → (⊤ → 𝑦 = 𝑧)))
32cbvalivw 2014 . . 3 (∀𝑥(⊤ → 𝑥 = 𝑧) → ∀𝑦(⊤ → 𝑦 = 𝑧))
43eximi 1836 . 2 (∃𝑧𝑥(⊤ → 𝑥 = 𝑧) → ∃𝑧𝑦(⊤ → 𝑦 = 𝑧))
5 df-mo 2601 . 2 (∃*𝑥⊤ ↔ ∃𝑧𝑥(⊤ → 𝑥 = 𝑧))
6 df-mo 2601 . 2 (∃*𝑦⊤ ↔ ∃𝑧𝑦(⊤ → 𝑦 = 𝑧))
74, 5, 63imtr4i 295 1 (∃*𝑥⊤ → ∃*𝑦⊤)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wtru 1539  wex 1781  ∃*wmo 2599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 210  df-ex 1782  df-mo 2601
This theorem is referenced by:  wl-motae  34919
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