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Mirrors > Home > MPE Home > Th. List > sbbib | Structured version Visualization version GIF version |
Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.) |
Ref | Expression |
---|---|
sbbib.y | ⊢ Ⅎ𝑦𝜑 |
sbbib.x | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
sbbib | ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1v 2157 | . . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
2 | sbbib.x | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | nfbi 1910 | . 2 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
4 | sbbib.y | . . 3 ⊢ Ⅎ𝑦𝜑 | |
5 | nfs1v 2157 | . . 3 ⊢ Ⅎ𝑦[𝑥 / 𝑦]𝜓 | |
6 | 4, 5 | nfbi 1910 | . 2 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑥 / 𝑦]𝜓) |
7 | sbequ12r 2249 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
8 | sbequ12 2248 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑦]𝜓)) | |
9 | 7, 8 | bibi12d 346 | . 2 ⊢ (𝑦 = 𝑥 → (([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ (𝜑 ↔ [𝑥 / 𝑦]𝜓))) |
10 | 3, 6, 9 | cbvalv1 2342 | 1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 Ⅎwnf 1790 [wsb 2071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 |
This theorem is referenced by: sbbibvv 2362 dfich2 44889 |
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