Theorem ich2ex 46808

Description: Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.)

Assertion

Ref	Expression
ich2ex	⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑

Proof of Theorem ich2ex

Step	Hyp	Ref	Expression
1		nfe1 2140	. 2 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
2		excom 2152	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2140	. . 3 ⊢ Ⅎ𝑦∃𝑦∃𝑥𝜑
4	2, 3	nfxfr 1848	. 2 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
5	1, 4	ichf 46790	1 ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑

Syntax hints:  ∃wex 1774  [wich 46785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167

This theorem depends on definitions:  df-bi 206  df-ex 1775  df-nf 1779  df-sb 2061  df-ich 46786

This theorem is referenced by: (None)