Theorem wl-moteq 37886

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.

Step	Hyp	Ref	Expression
1		dfmo 2544	. 2 ⊢ (∃*𝑥⊤ ↔ ∃𝑤∀𝑥(⊤ → 𝑥 = 𝑤))
2		stdpc5v 1945	. . . 4 ⊢ (∀𝑥(⊤ → 𝑥 = 𝑤) → (⊤ → ∀𝑥 𝑥 = 𝑤))
3		tru 1551	. . . . . 6 ⊢ ⊤
4	3	pm2.24i 150	. . . . 5 ⊢ (¬ ⊤ → 𝑦 = 𝑧)
5		aeveq 2065	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑤 → 𝑦 = 𝑧)
6	4, 5	ja 187	. . . 4 ⊢ ((⊤ → ∀𝑥 𝑥 = 𝑤) → 𝑦 = 𝑧)
7	2, 6	syl 17	. . 3 ⊢ (∀𝑥(⊤ → 𝑥 = 𝑤) → 𝑦 = 𝑧)
8	7	exlimiv 1937	. 2 ⊢ (∃𝑤∀𝑥(⊤ → 𝑥 = 𝑤) → 𝑦 = 𝑧)
9	1, 8	sylbi 218	1 ⊢ (∃*𝑥⊤ → 𝑦 = 𝑧)

Syntax hints:   → wi 4  ∀wal 1545  ⊤wtru 1548  ∃wex 1786  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-mo 2543

This theorem is referenced by:  wl-motae  37887