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| Mirrors > Home > MPE Home > Th. List > sbbibvv | Structured version Visualization version GIF version | ||
| Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023.) |
| Ref | Expression |
|---|---|
| sbbibvv | ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 1, 2 | sbbib 2364 | 1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 [wsb 2064 |
| 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 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: (None) |
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