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Mirrors > Home > MPE Home > Th. List > Mathboxes > ichnfb | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ichnfb | ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ichcom 44599 | . . . 4 ⊢ ([𝑥⇄𝑦]𝜑 ↔ [𝑦⇄𝑥]𝜑) | |
2 | ichnfim 44604 | . . . 4 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ [𝑦⇄𝑥]𝜑) → ∀𝑦Ⅎ𝑥𝜑) | |
3 | 1, 2 | sylan2b 597 | . . 3 ⊢ ((∀𝑥Ⅎ𝑦𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑦Ⅎ𝑥𝜑) |
4 | 3 | expcom 417 | . 2 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 → ∀𝑦Ⅎ𝑥𝜑)) |
5 | ichnfim 44604 | . . 3 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑) | |
6 | 5 | expcom 417 | . 2 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑦Ⅎ𝑥𝜑 → ∀𝑥Ⅎ𝑦𝜑)) |
7 | 4, 6 | impbid 215 | 1 ⊢ ([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 Ⅎwnf 1791 [wich 44585 |
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-5 1918 ax-6 1976 ax-7 2016 ax-10 2142 ax-11 2159 ax-12 2176 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2072 df-ich 44586 |
This theorem is referenced by: (None) |
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