Theorem ich2ex 48079

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 2186	. 2 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
2		excom 2198	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2186	. . 3 ⊢ Ⅎ𝑦∃𝑦∃𝑥𝜑
4	2, 3	nfxfr 1875	. 2 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
5	1, 4	ichf 48061	1 ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑

Syntax hints:  ∃wex 1801  [wich 48056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-11 2193  ax-12 2214

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-nf 1806  df-sb 2093  df-ich 48057

This theorem is referenced by: (None)