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Mirrors > Home > MPE Home > Th. List > sb9 | Structured version Visualization version GIF version |
Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2540. (Revised by Wolf Lammen, 15-Jun-2019.) |
Ref | Expression |
---|---|
sb9 | ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12a 2280 | . . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
2 | 1 | equcoms 2119 | . . . 4 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
3 | 2 | sps 2219 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
4 | 3 | dral1 2423 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
5 | nfnae 2418 | . . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
6 | nfnae 2418 | . . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
7 | nfsb2 2455 | . . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦[𝑥 / 𝑦]𝜑) | |
8 | 7 | naecoms 2413 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦[𝑥 / 𝑦]𝜑) |
9 | nfsb2 2455 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
10 | 2 | a1i 11 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))) |
11 | 5, 6, 8, 9, 10 | cbv2 2386 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)) |
12 | 4, 11 | pm2.61i 177 | 1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wal 1651 Ⅎwnf 1879 [wsb 2064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 |
This theorem is referenced by: sb9i 2540 |
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