Theorem wl-motae 34749

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 34747	. 2 ⊢ (∃*𝑢⊤ → ∃*𝑣⊤)
2		wl-moteq 34748	. . 3 ⊢ (∃*𝑣⊤ → 𝑦 = 𝑧)
3	2	alrimiv 1924	. 2 ⊢ (∃*𝑣⊤ → ∀𝑥 𝑦 = 𝑧)
4	1, 3	syl 17	1 ⊢ (∃*𝑢⊤ → ∀𝑥 𝑦 = 𝑧)

Syntax hints:   → wi 4  ∀wal 1531  ⊤wtru 1534  ∃*wmo 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1536  df-ex 1777  df-mo 2618

This theorem is referenced by:  wl-moae  34750