Theorem nfich1 48671

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 48670	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
2		nfa1 2190	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
3	1, 2	nfxfr 1885	1 ⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  Ⅎwnf 1815  [wsb 2100  [wich 48669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-10 2179

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816  df-ich 48670

This theorem is used by:  ichnfim  48688  ich2exprop  48695  ichreuopeq  48697