Theorem ich2ex 46682

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 2139	. 2 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
2		excom 2154	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2139	. . 3 ⊢ Ⅎ𝑦∃𝑦∃𝑥𝜑
4	2, 3	nfxfr 1847	. 2 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
5	1, 4	ichf 46664	1 ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑

Syntax hints:  ∃wex 1773  [wich 46659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778  df-sb 2060  df-ich 46660

This theorem is referenced by: (None)