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Theorem wl-moteq 34864
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-moteq (∃*𝑥⊤ → 𝑦 = 𝑧)

Proof of Theorem wl-moteq
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mo 2624 . 2 (∃*𝑥⊤ ↔ ∃𝑤𝑥(⊤ → 𝑥 = 𝑤))
2 stdpc5v 1940 . . . 4 (∀𝑥(⊤ → 𝑥 = 𝑤) → (⊤ → ∀𝑥 𝑥 = 𝑤))
3 tru 1542 . . . . . 6
43pm2.24i 153 . . . . 5 (¬ ⊤ → 𝑦 = 𝑧)
5 aeveq 2062 . . . . 5 (∀𝑥 𝑥 = 𝑤𝑦 = 𝑧)
64, 5ja 189 . . . 4 ((⊤ → ∀𝑥 𝑥 = 𝑤) → 𝑦 = 𝑧)
72, 6syl 17 . . 3 (∀𝑥(⊤ → 𝑥 = 𝑤) → 𝑦 = 𝑧)
87exlimiv 1932 . 2 (∃𝑤𝑥(⊤ → 𝑥 = 𝑤) → 𝑦 = 𝑧)
91, 8sylbi 220 1 (∃*𝑥⊤ → 𝑦 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wtru 1539  wex 1781  ∃*wmo 2622
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 1912  ax-6 1971  ax-7 2016
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-mo 2624
This theorem is referenced by:  wl-motae  34865
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