Theorem nfich2 47937

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 47935	. 2 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑))
2		nfa2 2188	. 2 ⊢ Ⅎ𝑦∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ 𝜑)
3	1, 2	nfxfr 1861	1 ⊢ Ⅎ𝑦[𝑥⇄𝑦]𝜑

Syntax hints:   ↔ wb 208  ∀wal 1546  Ⅎwnf 1791  [wsb 2074  [wich 47934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-10 2154  ax-11 2170

This theorem depends on definitions:  df-bi 209  df-or 855  df-ex 1788  df-nf 1792  df-ich 47935

This theorem is referenced by:  ich2exprop  47960  ichreuopeq  47962