Theorem ich2ex 47751

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 2156	. 2 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
2		excom 2168	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2156	. . 3 ⊢ Ⅎ𝑦∃𝑦∃𝑥𝜑
4	2, 3	nfxfr 1855	. 2 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
5	1, 4	ichf 47733	1 ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑

Syntax hints:  ∃wex 1781  [wich 47728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2183

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-sb 2069  df-ich 47729

This theorem is referenced by: (None)