Theorem wl-cbvmotv 34747

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.

Step	Hyp	Ref	Expression
1		ax7v2 2014	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
2	1	imim2d 57	. . . 4 ⊢ (𝑥 = 𝑦 → ((⊤ → 𝑥 = 𝑧) → (⊤ → 𝑦 = 𝑧)))
3	2	cbvalivw 2010	. . 3 ⊢ (∀𝑥(⊤ → 𝑥 = 𝑧) → ∀𝑦(⊤ → 𝑦 = 𝑧))
4	3	eximi 1831	. 2 ⊢ (∃𝑧∀𝑥(⊤ → 𝑥 = 𝑧) → ∃𝑧∀𝑦(⊤ → 𝑦 = 𝑧))
5		df-mo 2618	. 2 ⊢ (∃*𝑥⊤ ↔ ∃𝑧∀𝑥(⊤ → 𝑥 = 𝑧))
6		df-mo 2618	. 2 ⊢ (∃*𝑦⊤ ↔ ∃𝑧∀𝑦(⊤ → 𝑦 = 𝑧))
7	4, 5, 6	3imtr4i 294	1 ⊢ (∃*𝑥⊤ → ∃*𝑦⊤)

Syntax hints:   → wi 4  ∀wal 1531  ⊤wtru 1534  ∃wex 1776  ∃*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-ex 1777  df-mo 2618

This theorem is referenced by:  wl-motae  34749