Theorem ichnfb 48623

Description: If 𝑥 and 𝑦 are interchangeable in 𝜑, they are both free or both not free in 𝜑. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)

Assertion

Ref	Expression
ichnfb	⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem ichnfb

Step	Hyp	Ref	Expression
1		ichcom 48617	. . . 4 ⊢ ([𝑥⇄𝑦]𝜑 ↔ [𝑦⇄𝑥]𝜑)
2		ichnfim 48622	. . . 4 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ [𝑦⇄𝑥]𝜑) → ∀𝑦Ⅎ𝑥𝜑)
3	1, 2	sylan2b 606	. . 3 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑦Ⅎ𝑥𝜑)
4	3	expcom 419	. 2 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦Ⅎ𝑥𝜑))
5		ichnfim 48622	. . 3 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑)
6	5	expcom 419	. 2 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑦Ⅎ𝑥𝜑 → ∀𝑥Ⅎ𝑦𝜑))
7	4, 6	impbid 215	1 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  Ⅎwnf 1815  [wich 48603

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-ich 48604

This theorem is used by: (None)