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| Mirrors > Home > MPE Home > Th. List > sbcid | Structured version Visualization version GIF version | ||
| Description: An identity theorem for substitution. See sbid 2255. (Contributed by Mario Carneiro, 18-Feb-2017.) |
| Ref | Expression |
|---|---|
| sbcid | ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbsbc 3792 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑) | |
| 2 | sbid 2255 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
| 3 | 1, 2 | bitr3i 277 | 1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 [wsb 2064 [wsbc 3788 |
| 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-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: csbid 3912 snfil 23872 ex-natded9.26 30438 bnj605 34921 dedths 38963 frege93 43969 or2expropbilem1 47044 |
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