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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. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by BJ and WL, 9-Aug-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb2ae | ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drsb1 2501 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑢 / 𝑦][𝑣 / 𝑦]𝜑)) | |
2 | nfs1v 2157 | . . 3 ⊢ Ⅎ𝑦[𝑣 / 𝑦]𝜑 | |
3 | 2 | sbf 2267 | . 2 ⊢ ([𝑢 / 𝑦][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑) |
4 | 1, 3 | bitrdi 287 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 [wsb 2071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-10 2141 ax-12 2175 ax-13 2374 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1787 df-nf 1791 df-sb 2072 |
This theorem is referenced by: (None) |
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