Theorem ichnfb 43699

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 43691	. . . 4 ⊢ ([𝑥⇄𝑦]𝜑 ↔ [𝑦⇄𝑥]𝜑)
2		ichnfim 43698	. . . 4 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ [𝑦⇄𝑥]𝜑) → ∀𝑦Ⅎ𝑥𝜑)
3	1, 2	sylan2b 595	. . 3 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑦Ⅎ𝑥𝜑)
4	3	expcom 416	. 2 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦Ⅎ𝑥𝜑))
5		ichnfim 43698	. . 3 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑)
6	5	expcom 416	. 2 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑦Ⅎ𝑥𝜑 → ∀𝑥Ⅎ𝑦𝜑))
7	4, 6	impbid 214	1 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534  Ⅎwnf 1783  [wich 43679

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-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-ich 43680

This theorem is referenced by: (None)