Theorem ich2ex 48355

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 2187	. 2 ⊢ Ⅎ𝑥∃𝑥∃𝑦𝜑
2		excom 2199	. . 3 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
3		nfe1 2187	. . 3 ⊢ Ⅎ𝑦∃𝑦∃𝑥𝜑
4	2, 3	nfxfr 1886	. 2 ⊢ Ⅎ𝑦∃𝑥∃𝑦𝜑
5	1, 4	ichf 48337	1 ⊢ [𝑥⇄𝑦]∃𝑥∃𝑦𝜑

This proof depends on syntax axioms:  ∃wex 1812  [wich 48332

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-ich 48333

This theorem is used by: (None)