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Theorem wl-cbvmotv 33836
 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 2115 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
21imim2d 57 . . . 4 (𝑥 = 𝑦 → ((⊤ → 𝑥 = 𝑧) → (⊤ → 𝑦 = 𝑧)))
32cbvalivw 2111 . . 3 (∀𝑥(⊤ → 𝑥 = 𝑧) → ∀𝑦(⊤ → 𝑦 = 𝑧))
43eximi 1933 . 2 (∃𝑧𝑥(⊤ → 𝑥 = 𝑧) → ∃𝑧𝑦(⊤ → 𝑦 = 𝑧))
5 df-mo 2605 . 2 (∃*𝑥⊤ ↔ ∃𝑧𝑥(⊤ → 𝑥 = 𝑧))
6 df-mo 2605 . 2 (∃*𝑦⊤ ↔ ∃𝑧𝑦(⊤ → 𝑦 = 𝑧))
74, 5, 63imtr4i 284 1 (∃*𝑥⊤ → ∃*𝑦⊤)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1654  ⊤wtru 1657  ∃wex 1878  ∃*wmo 2603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112 This theorem depends on definitions:  df-bi 199  df-ex 1879  df-mo 2605 This theorem is referenced by:  wl-motae  33838
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