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| Mirrors > Home > MPE Home > Th. List > cbvsbv | Structured version Visualization version GIF version | ||
| Description: Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was extracted from a former cbvabv 2812 version. (Contributed by Wolf Lammen, 16-Mar-2025.) |
| Ref | Expression |
|---|---|
| cbvsbv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| cbvsbv | ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbco2vv 2099 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
| 2 | cbvsbv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | sbievw 2093 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
| 4 | 3 | sbbii 2076 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
| 5 | 1, 4 | bitr3i 277 | 1 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 [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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 |
| This theorem is referenced by: sbco4lem 2101 cbvabv 2812 |
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