Theorem wl-motae 38198

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-motae	⊢ (∃*𝑢⊤ → ∀𝑥 𝑦 = 𝑧)

Proof of Theorem wl-motae

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wl-cbvmotv 38196	. 2 ⊢ (∃*𝑢⊤ → ∃*𝑣⊤)
2		wl-moteq 38197	. . 3 ⊢ (∃*𝑣⊤ → 𝑦 = 𝑧)
3	2	alrimiv 1956	. 2 ⊢ (∃*𝑣⊤ → ∀𝑥 𝑦 = 𝑧)
4	1, 3	syl 18	1 ⊢ (∃*𝑢⊤ → ∀𝑥 𝑦 = 𝑧)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ⊤wtru 1570  ∃*wmo 2564

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-mo 2566

This theorem is used by:  wl-moae  38199