Theorem nfich1 43682

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 43681	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
2		nfa1 2154	. 2 ⊢ Ⅎ𝑥∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
3	1, 2	nfxfr 1852	1 ⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 208  ∀wal 1534  Ⅎwnf 1783  [wsb 2068  [wich 43680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-10 2144

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1780  df-nf 1784  df-ich 43681

This theorem is referenced by:  ichnfim  43694  ich2exprop  43703  ichreuopeq  43705