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Mirrors > Home > MPE Home > Th. List > sb2ae | Structured version Visualization version GIF version |
Description: In the case of two successive substitutions for two always equal variables, the second substitution has no effect. (Contributed by BJ and WL, 9-Aug-2023.) |
Ref | Expression |
---|---|
sb2ae | ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drsb1 2489 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑦][𝑣 / 𝑦]𝜑)) | |
2 | nfs1v 2237 | . . 3 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑 | |
3 | 2 | sbf 2234 | . 2 ⊢ ([𝑢 / 𝑦][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑) |
4 | 1, 3 | syl6bb 288 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1520 [wsb 2042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-10 2112 ax-12 2141 ax-13 2344 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1762 df-nf 1766 df-sb 2043 |
This theorem is referenced by: (None) |
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