Theorem nfich2 47812

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

Assertion

Ref	Expression
nfich2	⊢ Ⅎ𝑦[𝑥⇄𝑦]𝜑

Proof of Theorem nfich2

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ich 47810	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
2		nfa2 2182	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
3	1, 2	nfxfr 1855	1 ⊢ Ⅎ𝑦[𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 206  ∀wal 1540  Ⅎwnf 1785  [wsb 2068  [wich 47809

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-10 2147  ax-11 2163

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1782  df-nf 1786  df-ich 47810

This theorem is referenced by:  ich2exprop  47835  ichreuopeq  47837