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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 2810 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 2097 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
2 | cbvsbv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbievw 2091 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
4 | 3 | sbbii 2074 | . 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 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 |
This theorem is referenced by: sbco4lem 2099 cbvabv 2810 |
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