Theorem ich2ex 48141

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 2191	. 2 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
2		excom 2203	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2191	. . 3 ⊢ Ⅎ𝑦∃𝑦∃𝑥𝜑
4	2, 3	nfxfr 1880	. 2 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
5	1, 4	ichf 48123	1 ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑

Syntax hints:  ∃wex 1806  [wich 48118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-nf 1811  df-sb 2098  df-ich 48119

This theorem is referenced by: (None)