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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 2246. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Ref | Expression |
---|---|
sbcid | ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbsbc 3781 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑) | |
2 | sbid 2246 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
3 | 1, 2 | bitr3i 277 | 1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2066 [wsbc 3777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-sbc 3778 |
This theorem is referenced by: csbid 3906 snfil 23688 ex-natded9.26 30105 bnj605 34382 dedths 38296 frege93 43170 or2expropbilem1 46201 |
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