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| Mirrors > Home > MPE Home > Th. List > sb6rfv | Structured version Visualization version GIF version | ||
| Description: Reversed substitution. Version of sb6rf 2473 requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993.) (Revised by Wolf Lammen, 7-Feb-2023.) |
| Ref | Expression |
|---|---|
| sb6rfv.nf | ⊢ Ⅎ𝑦𝜑 |
| Ref | Expression |
|---|---|
| sb6rfv | ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sb6rfv.nf | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 2 | sbequ12r 2252 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
| 3 | 1, 2 | equsalv 2267 | . 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑) |
| 4 | 3 | bicomi 224 | 1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 Ⅎwnf 1783 [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-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: eu1 2610 |
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