Theorem nfich1 48216

Description: The first interchangeable setvar variable is not free. (Contributed by AV, 21-Aug-2023.)

Assertion

Ref	Expression
nfich1	⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑

Proof of Theorem nfich1

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ich 48215	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
2		nfa1 2186	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
3	1, 2	nfxfr 1883	1 ⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 209  ∀wal 1568  Ⅎwnf 1813  [wsb 2096  [wich 48214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-10 2176

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1810  df-nf 1814  df-ich 48215

This theorem is referenced by:  ichnfim  48233  ich2exprop  48240  ichreuopeq  48242